Vol.4,    Part?  DECEMBER,  1922  ,     .  Number  25 


BULLETIN 


OF  THE 


NATIONAL  RESEARCH 
COUNCIL 


CELESTIAL  MECHANICS 


A  Survey  of  the  Status  of  the  Determination  of  the  General 
Perturbations  of  the  Minor  Planets 

BY 
A.  0.  LEUSCHNER 


PUBLISHED  BY  THE  NATIONAL  RESEARCH  COUNCIL 

OP 

THE  NATIONAL  ACADEMY  OP  SCIENCES 

WASHINGTON,  D.  C, 

1922 


Announcement  Concerning  Publications 

of  the 

National  Research  Council 


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BULLETIN 

OF  THE 

NATIONAL  RESEARCH  COUNCIL 

Vol.  4,  Part  7.  December,  1922  Number  25 

CELESTIAL  MECHANICS 

A  SURVEY  OF  THE  STATUS  OF  THE  DETERMINATION  OF 

THE  GENERAL  PERTURBATIONS  OF  THE 

MINOR  PLANETS. 

Appendix  to  the  Report  of  the  Committee  on  Celestial  Mechanics, 
National  Research  Council* 

BY  A.  0.  LEUSCHNER 
Professor  of  Astronomy,  University  of  California 


CONTENTS 

PAGE 

Introduction 2 

1  Ceres 11 

2  Pallas 15 

3  Juno , 20 

4  Vesta 23 

10  Hygiea 28 

—    28  Bellona 31 

93  Minerva 35 

94  Aurora 38 

127  Johanna 40 

128  Nemesis 42 

175  Andromache 44 

433  Eros 46 

447  Valentine 51 

588  Achilles 54 

617  Patroclus 57 

624  Hector 60 

659  Nestor 63 

716  Berkeley 66 

718  Erida 68 

884  Priamus 70 

911  Agamemnon 73 

*This  Committee  of  the  Division  of  Physical  Sciences  of  the  National  Re- 
search Council  consists  of  the  following  members:  E.  W.  Brown,  Professor  of 
Mathematics,  Yale  University,  Chairman;  G.  D.  Birkhoff,  Professor  of  Mathe- 
matics, Harvard  University ;  A.  O.  Leuschner,  Professor  of  Astronomy,  University 
of  California;  H.  N.  Russell,  Professor  of  Astronomy,  Princeton  University, 


670869 


A/3 

{•     ' 


INTRODUCTION. 

With  approximately  one  thousand  asteroids  discovered  and  believed 
to  be  sufficiently  observed  to  permit  of  fairly  reliable  orbit  determina- 
tions, as  indicated  by  the  permanent  numbers  assigned  to  them,  the 
task  of  preserving  these  discoveries  has  grown  so  stupendous  that  the 
time  seems  to  have  arrived  for  an  analysis  of  the  present  astronomical 
practice  in  providing  the  necessary  additional  observations  and 
calculations. 

Hitherto,  the  burden  of  correcting  orbit  elements  and  computing 
ephemerides  has  rested  principally  on  the  Berlin  Recheninstitut.  In 
recent  years  the  Marseilles  Observatory  has  rendered  notable  service 
in  contributing  orbits  and  ephemerides.  Observations,  photographic 
and  visual,  are  regularly  made  at  a  number  of  observatories.  The 
Berlin  Recheninstitut  publishes  ephemerides  and  other  results  in  the 
Astronomische  Nachrichten,  and  in  the  Ephemeriden  der  Kleinen 
Planeten.  Up  to  1918  these  data  appeared  also  in  the  Astronomisches 
Jahrbuch.  The  number  of  oppositions  during  which  the  minor  planets 
have  been  observed,  and  the  status  of  orbit  determinations  are  an- 
nually summarized  in  the  Viertelj  ahrschrift  der  Astronomischen  Gesell- 
schaft.  In  1901  Bauschinger  published  the  latest  reliable  elements,  etc., 
with  data  concerning  the  perturbations  for  the  then  known  463  planets, 
"Tabellen  zur  Geschichte  und  Statistik  der  Kleinen  Planeten." 

The  latest  available  collection  of  elements  is  contained  in  the 
Berlin  Jahrbuch  for  1918.  The  adopted  Jahrbuch  elements  serve  the 
purpose  of  providing  ephemerides  from  opposition  to  opposition. 
Their  origin  may  be  traced  from  the  notes  given  from  year  to  year 
in  the  Jahrbuch,  ending  with  1918,  and  in  Kleine  Planeten.  Some  of 
the  elements  include  arbitrary  corrections  to  the  mean  motion  and 
to  the  mean  anomaly  for  the  purpose  of  representing  late  oppositions 
so  as  to  serve  for  prediction  of  immediately  following  oppositions. 
In  other  cases,  approximate  or  accurate  perturbations  are  included, 
with  or  without  correction  of  the  elements  by  the  usual  least  squares 
adjustment.  For  thirty-six  planets  the  elements  in  the  Jahrbuch  of 
1918  are  mean  or  osculating  elements,  derived  in  connection  with 
general  perturbations  which  are  approximately  included  in  the  pre- 
diction of  ephemerides.  Until  similar  fundamental  data  shall  have 
become  available  for  the  remaining  planets,  the  present  practice  of 
the  Recheninstitut  appears  to  furnish  the  only  certain  method  for 
the  preservation  of  planetary  discoveries. 


CELESTIAL   MECHANICS:    LEUSCHNER  3 

In  addition  to  the  general  pertubations  of  the  thirty-six  planets 
which  are  being  used  by  the  Jahrbuch,  the  general  pertubations  of  a 
number  of  other  planets  have  been  .derived  on  the  basis  of  what,  at 
the  time,  appeared  to  be  reliable  osculating  elements.  These  pre- 
sumably valuable  data  have  been  replaced  by  later  elements  derived 
independently,  more  or  less  accurately,  with  or  without  general 
perturbations,  or  upon  the  basis  of  arbitrary  corrections.  Bausch- 
inger's  Tabellen  form  a  valuable  key  to  some  of  these  investigations, 
but  even  in  the  Tabellen  the  elements  and  perturbations  cited  do  not, 
in  all  cases,  represent  the  best  elements  and  perturbations  available, 
although  perhaps  in  every  case  they  are  the  most  reliable  for  subse- 
quent oppositions.  This  arises  from  the  fact  that  earlier  investigations 
were  abandoned  by  Bauschinger  in  favor  of  later  ones.  Preliminary 
calculations  have  shown,  however,  that  some  of  the  earlier  elements 
and  perturbations  represent  distant  oppositions  at  a  later  date  more 
satisfactorily  than  his  adopted  elements  and  perturbations  represent 
earlier  oppositions  equally  remote. 

Of  great  importance  for  the  program  of  the  Recheninstitut  are  the 
contributions  of  Brendel,  who  has  developed  methods  for  the  approxi- 
mate determination  of  the  perturbations  for  certain  groups  of  planets. 
Perturbations  greater  than  3'.4  within  fifty  years  are  included,  with 
the  object  of  reproducing  geocentric  places  within  20'  for  100  years. 
So  far  the  necessary  data  have  been  published  by  Brendel,  Labitzke, 
and  Boda  for  230  planets,  approximately  25  per  cent  of  the  total 
number  of  known  minor  planets.  The  advantage  to  be  gained  from 
BrendeFs  contributions  for  these  planets  is  that  for  the  practical 
purpose  of  preserving  these  planets,  by  following  their  motion,  it 
should  become  unnecessary  as  a  rule  to  compute  special  perturbations 
for  them,  or  even  to  apply  corrections  to  the  elements.  Brendel  plans 
to  continue  the  work  of  supplying  instantaneous  elements  and 
approximate  perturbations  for  other  groups  of  planets  so  that  the 
program  of  the  Recheninstitut,  of  the  Marseilles  Observatory,  and  of 
various  investigators  who,  from  time  to  time,  publish  improved 
elements  and  perturbations  for  ephemeris  purposes,  will  become 
more  and  more  simplified. 

The  preservation  of  planetary  discoveries  by  observation  and  pre- 
diction with  the  aid  of  approximate  perturbations  is  not  the  ultimate 
aim  of  astronomical  science,  but  a  necessary  and  unavoidable  means 
to  the  end.  The  ultimate  aim  rests  on  the  determination  of  mean 
elements  and  general  perturbations  which  hold  for  all  time  or  at  least 
for  very  long  periods  within  the  limits  of  accuracy  set  by  observation. 
It  is  expected  that  the  elements  and  perturbations  determined  under 


4  CELESTIAL   MECHANICS:    LEUSCHNER 

the  Newtonian  law  of  gravitation  may  serve  this  purpose,  provided 
that  the  mathematical  difficulties  will  not  prove  insurmountable.  It 
may  be  assumed  that  the  rigid  mathematical  methods  hitherto 
developed  are  satisfactory  for  planets  with  moderate  eccentricity  and 
inclination  which  are  not  in  a  very  near  commensurable  ratio  with 
any  of  the  major  planets,  but  it  has  not  been  established  so  far 
whether  an  accurate  application  of  the  Newtonian  law  would  fully 
account  for  the  motion  of  the  minor  planets  even  in  the  ordinary  cases 
just  referred  to. 

Exhaustive  researches  are  available  only  for  a  very  limited  number 
of  planets.  Among  these  are  (4)  Vesta,  (13)  Egeria  and  (447)  Valen- 
tine. The  researches  on  (4)  Vesta  are  due  to  Leveau,  whose  extraordi- 
nary investigations  extend  approximately  over  a  complete  century  of 
oppositions.  In  connection  with  his  work  on  the  motion  of  Vesta, 
Leveau  has  aimed  at  a  determination  of  the  masses  of  Jupiter  and 
Mars.  His  final  value  is  larger  than  the  best  available  mass 
of  Jupiter  by  approximately  one  one-thousandth.  On  account 
of  the  moderate  perturbations,  the  motion  of  Vesta  does  not 
lend  itself  as  well  to  a  determination  of  the  mass  of  Jupiter 
as  the  motion  of  other  minor  planets  with  very  large  perturba- 
tions. Any  slight  departure  from  the  true  mass  of  Jupiter, 
et  cetera,  can  reveal  itself  through  the  motion  of  Vesta  only  in  long 
intervals  of  time,  which  accounts  for  Leveau's  gradual  improvement 
of  his  adopted  mass  by  successively  including  longer  periods  of 
observation.  For  the  present  his  results  may  be  considered  funda- 
mental and  final,  so  far  as  this  planet  is  concerned.  No  other  case  has 
been  studied  so  exhaustively.  Later  predictions  are  well  within  the 
errors  of  observation,  and  not  the  slightest  departure  from  the  New- 
tonian law  is  noticeable.  It  remains,  however,  to  establish  the  same 
result  for  planets  with  large  perturbations,  particularly  for  such 
planets  as  have  a  mean  motion  commensurable  with  that  of  Jupiter. 
To  avoid  the  necessity  of  gradually  improving  the  Jupiter  mass  by 
means  of  subsequent  observations  of  Vesta,  it  appears  advisable  to 
base  further  predictions  on  the  best  determined  values  of  the  masses 
of  the  major  planets.  Vesta  also  furnishes  an  example  of  the  weight 
to  be  assigned  to  observations  in  the  early  part  of  the  last  century. 

Leveau's  investigations  furnish  a  striking  example  of  the  funda- 
mental researches  necessary  for  the  promotion  of  astronomical  science 
as  distinguished  from  the  generally  accepted  program  of  observation 
and  prediction  for  the  preservation  of  discoveries. 

For  the  study  and  interpretation  of  planetary  statistics,  particularly 
with  reference  to  the  origin  of  minor  planets,  the  explanation  of  the 


CELESTIAL   MECHANICS:    LEUSCHNER  5 

gaps,  the  question  of  stability  and  ultimate  destiny,  or  in  general 
regarding  their  place  in  any  hypothesis  concerning  the  solar  system, 
final  mean  elements  derived  on  the  basis  of  accurate  developments  of 
the  perturbations  are  most  essential.  Fragments  of  fundamental 
investigations  of  perturbations  are  available  for  a  number  of  minor 
planets.  The  value  of  some  of  these  has  been  vitiated  by  corrections 
made  in  connection  with  the  accepted  program  of  approximate  pre- 
diction, such  as,  for  example,  the  correction  of  an  accurate  set  of 
osculating  elements  derived  by  special  or  general  perturbations,  to 
represent  later  oppositions,  either  without  perturbations  or  by  taking 
account  only  of  approximate  or  incomplete  perturbations. 

Fundamental  investigations  here  are  understood  to  include  the 
determination  of  osculating  or  mean  elements  from  a  limited  number 
of  oppositions  with  complete  regard  of  the  perturbations,  either  special 
or  general,  in  so  far  as  they  may  have  been  appreciable.  In  con- 
nection with  the  study  of  the  data  existing  for  a  limited  number  of 
selected  planets,  it  has  been  found  that  the  failure  of  such  elements 
and  perturbations  to  represent  future  oppositions  in  some  cases  can 
be  accounted  for  by  the  fact  that  the  masses  of  the  major  planets 
were  known  at  the  time  with  insufficient  accuracy.  The  mere  cor- 
rection of  the  perturbations  therefore,  for  the  latest  known  values  of 
the  masses  may  render  such  elements  and  perturbations  far  more 
satisfactory  than  they  appeared  to  be  at  the  time  when  they  were 
discarded  in  favor  of  new  determinations  of  elements  with  or  without 
perturbations. 

Freed  from  effects  of  changes  which  affect  disadvantageously  their 
permanent  value  the  fragments  of  fundamental  investigations  referred 
to  are  of  great  importance  as  a  basis  for  researches  and  their  intel- 
ligent application  will  involve  a  vast  saving  in  computational  and 
theoretical  work. 

At  present  it  appears  next  to  hopeless  to  the  investigator  to  adopt 
a  profitable  form  of  attack  in  connection  with  any  of  the  older  minor 
planets  without  an  enormous  expenditure  of  time  in  searching  astro- 
nomical records.  This  accounts  for  the  many  duplications  of  effort 
and  for  the  disregard  of  previous  valuable  investigations.  If 
systematically  undertaken,  the  task  of  bringing  to  light  the  important 
data  available  for  a  final  determination  of  the  elements  and  general 
perturbations  of  the  minor  planets,  does  not  appear  insurmountable. 
Once  available,  such  research  surveys  will  be  invaluable  and 
should  prove  an  encouragement  to  research,  particularly  to  young 
investigators. 

The  research  surveys  of  the  few  planets  which  are  given  below  are 


6  CELESTIAL   MECHANICS:   LEUSCHNER 

intended  to  serve  as  illustrations  of  the  data  which  should  be  made 
easily  accessible.  No  claim  is  made  for  the  absolute  completeness  of 
these  data.  The  time  for  active  work,  with  the  aid  of  a  few  assistants, 
to  prepare  these  preliminary  surveys  has  extended  only  over  a  little 
more  than  a  month.  A  great  mass  of  material  had  to  be  consulted 
which  was  found  to  be  of  no  importance  to  the  purpose  in  hand..  This 
is  being  preserved  on  cards  for  easy  reference,  if  required  at  any  time. 
Thus  care  has  been  taken  to  eliminate  elements  which  would  not  be 
considered  as  fairly  accurate  osculating  elements  particularly  those 
which  have  resulted  from  corrections  on  the  basis  of  subsequent 
oppositions  purely  for  ephemeris  purposes,  without  complete  con- 
sideration of  the  perturbations  and  of  the  earlier  oppositions  in  the 
final  adjustment.  This  policy,  however,  has  not  been  adhered  to 
strictly,  partly  for  historical  and  theoretical  reasons  with  reference  to 
preliminary  elements,  and  partly  for  other  reasons  with  reference  to 
later  elements. 

Whenever  possible,  the  reasons  for  the  abandonment  of  previous 
investigations  are  given,  but  in  many  cases  no  reasons  could  be  found, 
at  least  not  in  the  astronomical  records  available  in  the  library  of  the 
University  of  California.  Some  of  these  reasons  are  probably  to  be 
found  in  the  records  available  in  the  library  of  the  Lick  Observatory, 
but  in  the  limited  time  it  has  not  been  possible  to  consult  these  or 
other  additional  records  for  this  preliminary  survey.  An  immense 
amount  of  fundamental  work  has  been  accomplished  by  the  Berlin 
Recheninstitut,  particularly  in  computing  special  perturbations  and 
deriving  osculating  elements,  but  has  been  published  only  in  part. 
The  remainder  reposes  in  the  archives  of  the  Recheninstitut.  It  may 
be  assumed  that  the  immense  task  of  providing  ephemerides  has 
interfered  with  the  publication  of  the  accumulated  material.  With- 
out this  material,  research  surveys  such  as  those  presented  here  are 
not  complete. 

A  simple  way  of  accomplishing  the  introduction  of  the  improved 
mass  of  disturbing  planets  referred  to  above,  is  to  multiply  the  final 
sum  of  all  the  terms  for  each  component  of  the  perturbations  by  the 
ratio  of  the  new  to  the  old  mass.  Aside  from  the  improvement  which  it 
may  be  possible  to  make  to  some  of  the  older  fundamental  data,  par- 
ticularly those  which  are  no  longer  used  for  ephemeris  purposes,  by 
the  introduction  of  the  best  determined  values  of  the  masses  of  the 
major  planets,  it  is  probably  possible  to  enhance  their  values  still 
further  by  correcting  the  elements  on  the  basis  of  the  existing  develop- 
ments of  the  general  perturbations,  with  the  aid  of  subsequent 
oppositions  and  in  case  of  appreciable  changes  in  the  elements,  by 


CELESTIAL   MECHANICS:    LEUSCHNER  7 

also  correcting  the  numerical  coefficients  in  the  general  developments 
by  differential  methods. 

After  revision  of  independently  determined  elements  and  pertur- 
bations for  separate  series  of  fairly  consecutive  oppositions,  discon- 
nected by  a  gap  including  a  number  of  oppositions  to  which  neither 
series  was  extended  either  backward  or  forward,  the  separate  funda- 
mental investigations  will,  in  some  cases,  probably  be  found  to  be 
entirely  consistent  and  thus  become  of  permanent  value,  such  as 
Leveau's  investigations  on  (4)  Vesta,  without  involving  extensive 
theoretical  and  numerical  work.  In  other  similar  cases  the  correction 
of  the  elements  and  perturbations  pertaining  to  fundamental  investi- 
gations for  groups  of  oppositions  separated  by  considerable  gaps,  so 
as  to  represent  the  osculating  data  at  a  subsequent  epoch,  may  establish 
satisfactorily  the  connection  between  one  or  more  groups  of  oppositions 
for  which  elements  and  perturbations  have  been  independently  deter- 
mined with  accuracy.  The  mode  of  attack  will  vary  with  the  avail- 
able data  for  different  planets,  as  indicated  by  the  research  surveys, 
which  this  discussion  advocates.  The  resurrection  of  the  classical 
contributions  of  the  pioneer  investigators  of  planetary  perturbations 
on  a  permanent  basis,  should  produce  material  of  great  value 
for  the  ultimate  aims  of  astronomical  science  concerning  planetary 
investigations. 

The  proposed  program  of  fundamental  investigations  cannot 
supersede  the  present  astronomical  practice  in  caring  for  the  minor 
planets  in  the  immediate  future,  but  as  stated  above  it  will  be  of 
great  assistance  for  the  practical  purposes  of  prediction,  and  should 
gradually  solve  the  now  stupendous  task  of  preserving  planetary 
discoveries,  while  furnishing  at  the  same  time  the  data  for  the  more 
fundamental  aims  of  astronomical  science. 

For  the  majority  of  the  minor  planets,  probably  the  application 
of  four  successive  steps  or  processes  will  be  necessary  to  preserve  the 
discoveries  until  final  elements  and  perturbations  can  be  made  avail- 
able. The  first  step  or  process  represents  the  present  practice 
principally  conducted  by  the  Berlin  Recheninstitut.  The  second  step 
is  illustrated  by  Brendel's  plan  of  supplying  instantaneous  elements 
and  approximate  perturbations.  The  third  step  corresponds  to  the 
determination  of  the  elements  and  perturbations  of  the  Watson 
asteroids  undertaken  by  Leuschner,  which  are  intended  to  provide 
fairly  accurate  but  not  final  results.  Hansen's  and  the  Bohlin-v. 
Zeipel  methods  have  been  found  most  practical  and  accurate  in  this 
connection.  The  fourth  and  final  step  is  demonstrated  by  the  funda- 
mental work  of  Leveau  on  (4)  Vesta.  It  is  the  object  of  this  discus- 


8  CELESTIAL  MECHANICS:   LEVSCHNER 

sion  to  encourage  researches  similar  to  Leveau's,  and  by  supplying 
samples  of  research  surveys  for  a  limited  number  of  planets  to  pave 
the  way  for  a  comprehensive  international  program  in  this  connection. 

It  cannot  be  too  strongly  emphasized  that  accurate  osculating 
elements  are  absolutely  essential  for  fundamental  investigations  of 
the  perturbations.  While  this  requirement  is  fully  recognized,  the 
prevailing  practice  of  changing  elements  for  immediate  ephemeris 
purposes  is  apt  to  lead  to  erroneous  interpretation  of  available 
elements.  Mean  elements,  in  general,  can  be  determined  only  after 
osculating  elements  and  perturbations  shall  have  become  available. 
Some  investigators  have  adopted  as  approximate  mean  elements  the 
average  of  elements  published  for  more  or  less  extensive  series  of 
oppositions,  assuming  that  these  elements  represent  fairly  reliable 
osculating  elements.  Even  if  this  were  the  case,  it  hardly  ever  occurs 
that  a  sufficiently  large  number  of  elements,  uniformly  distributed 
over  the  orbit,  are  available  to  guarantee  that,  in  taking  the  average, 
the  effect  of  the  periodic  terms  is  entirely  eliminated.  But,  as 
previously  stated,  many  of  the  apparently  reliable  sets  of  elements 
are  not  osculating,  but  inferior  elements  produced  by  arbitrary 
changes  or  with  incomplete  perturbations. 

Practically  the  only  reliable  method  of  arriving  at  accurate  initial 
osculating  elements  consists  in  representing  the  observations  of  a 
limited  number  of  oppositions  by  taking  into  account  the  special 
perturbations  and  in  testing  the  validity  of  the  resulting  elements  for 
one  or  more  oppositions  following.  Osculating  elements  thus  obtained 
will  rarely  require  later  changes  which  would  affect  the  coefficients 
of  the  general  perturbations.  No  correction  of  such  elements  should 
be  attempted,  except  on  the  basis  of  the  determination  of  complete 
special  or  general  perturbations.  As  it  was  not  considered  necessary, 
at  the  time,  to  adhere  strictly  to  the  foregoing  principle  in 
Leuschner's  program  for  the  determination  of  the  perturbations  of 
Watson's  asteroids,  allowances  for  slight  inaccuracies  may  later 
become  necessary  for  some  of  the  Watson  planets.  In  particular, 
corrections  to  the  larger  coefficients  of  the  perturbations  may  be 
necessary  for  the  planets  for  which  the  initial  adopted  elements,  con- 
sidered at  the  time  as  sufficiently  accurate,  were  neither  accurate 
mean  elements  nor  accurate  osculating  elements. 

Attention  has  recently  been  called,  in  the  Proceedings  of  the 
National  Academy  of  Sciences  of  1921,  Vol.  8,  No.  7,  p.  170,  and  in 
the  report  of  the  Committee  on  Celestial  Mechanics  of  the  National 
Research  Council,  Bulletin  of  the  National  Research  Council, 
Vol.  3,  Part  4,  No.  19,  June,  1922,  to  the  extremely  satis- 


CELESTIAL   MECHANICS:   LEUSCHNER  0 

factory  results  obtained  for  the  planets  (10)  Hygiea,  and  (175) 
Andromache,  by  the  application  of  Leuschner's  revision  of  von 
ZeipePs  tables  for  the  Hecuba  group.  Further  reference  to  the  great 
importance  of  "Gruppenweise  Berechnung  der  Stoerungen,"  inaug- 
urated by  Bohlin,  may  therefore  be  omitted  here.  The  methods  of 
Bohlin  and  his  followers  serve  admirably  in  connection  with  the  third 
of  the  four  stages  outlined  above  for  the  determination  of  funda- 
mental results.  In  certain  cases  of  limited  eccentricity  and  inclination, 
they  will,  no  doubt,  lead  to  final  results. 

No  claim  is  made  that  the  planets  for  which  research  surveys  are 
given  below  are  the  ones  most  in  need  of  immediate  attention. 
Further  study  of  available  data  will  be  necessary  to  classify  the 
planets  with  reference  to  the  requirements  of  observation  and  com- 
putation, as  outlined  in  the  report  of  the  American  Committee  on 
Comets  and  Asteroids,  presented  at  the  Brussels  meeting  of  the 
International  Astronomical  Union  in  1919;  nor  are  the  planets  con- 
sidered below  the  most  important  for  fundamental  scientific  purposes. 
The  list,  however,  may  be  considered  as  fairly  representative  of  the 
immediate  research  requirements.  To  some  extent  the  selection  has 
been  accidental.  Thus  the  computing  section  of  the  British  Astro- 
nomical Society  has  undertaken  the  computation  of  the  ephemerides 
of  the  first  four  planets.  In  this  connection  it  sought  advice  regarding 
the  best  available  data  and  methods  of  procedure.  The  research  sur- 
veys of  the  first  four  planets  were  undertaken  to  aid  the  computing 
section  in  its  undertaking.  The  importance  of  the  Trojan  group  is 
too  well  known  to  be  emphasized.  For  further  investigations  con- 
cerning the  theories  of  the  six  planets  belonging  to  this  group  the 
research  surveys  given  will  be  of  considerable  value.  It  is  of  interest 
to  note  that  Leuschner's  orbit  methods  as  applied  by  Einarsson, 
appear  to  be  the  most  promising  for  the  determination  of  preliminary 
osculating  elements,  while  Wilkens'  method  deserves  careful  trial 
in  deriving  the  perturbations.  E.  W.  Brown's  unpublished  theory 
promises  to  be  thoroughly  fundamental.  For  the  two  planets  of  the 
Trojan  group  last  discovered,  more  accurate  preliminary  osculating 
elements  are  immediately  needed.  For  other  planets,  the  list  of 
research  surveys  themselves  will  reveal  the  most  necessary  work  to 
be  done.  In  general,  reference  to  theoretical  investigations  is  included 
only  in  connection  with  a  simultaneous  new  determination  of  elements. 
Thus  the  numerous  and  important  investigations  on  the  theory  of 
the  Trojan  group  are  not  considered  here,  the  chief  object  of  the 
surveys  of  these  planets  being  to  furnish  numerical  data  and  encourage 
their  improvement  as  a  basis  for  such  theories. 


10  CELESTIAL   MECHANICS:   LEUSCHNER 

The  form  in  which  the  research  surveys  are  presented  must  be  con- 
sidered experimental.  That  adopted  is  the  outcome  of  several  other 
attempts  at  presenting  the  material.  It  is  hoped  that  this  report  will 
call  forth  helpful  criticisms  and  suggestions  which  may  ultimately 
lead  to  the  adoption  of  some  definite  plan  for  international  coopera- 
tion. Much  material  has  been  collected  on  planets  not  included  in  the 
list,  which,  it  is  hoped,  may  be  printed  later. 

It  was  found  that  the  research  surveys  for  the  various  planets  could 
not  be  made  so  complete  that  the  investigator  may  abstain  from 
referring  to  the  sources  themselves.  This  applies  also  to  the  collection 
of  elements.  The  elements  are  collected  merely  for  purposes  of  com- 
parison and  are  not  reproduced  with  uniform  accuracy. 

For  practically  all  the  planets  in  the  list,  except  the  first  four  and 
several  others,  a  fairly  complete  bibliography  of  observations  has 
been  prepared,  but  this  bibliography  is  published  here  only  for  the 
last  two  of  the  Trojan  group. 

Attention  might  well  be  called  here  to  the  need  of  curtailing 
indiscriminate  observations.  Even  in  recent  years  observations  have 
been  multiplied  for  planets  for  which  two  or  three  accurate  observa- 
tions at  each  opposition  would  be  sufficient  for  all  scientific  purposes. 
It  is  planned  to  formulate  in  the  near  future  definite  proposals  for 
an  international  program  of  observations. 

The  main  purpose  of  this  report  is  the  encouragement  of  funda- 
mental researches  essential  to  the  ultimate  aims  of  astronomical 
science,  which,  for  their  consummation,  require  the  knowledge  of 
accurate  elements  and  perturbations  of  the  minor  planets. 

The  surveys  have  been  prepared  in  the  main  by  Dr.  W.  F.  Meyer, 
and  by  Dr.  H.  Thiele,  assisted  by  several  advanced  students  in 
astronomy,  who  have  gathered  the  necessary  references.  For  the 
Trojan  group,  unpublished  data  collected  by  Dr.  Sturla  Einarsson 
have  been  available. 

As  a  rule  the  abbreviations  adopted  for  the  references  are  those  of 
the  Astronomischer  Jahresbericht. 

The  usual  notations  of  the  elements  are  adhered  to,  both  ^  and  n 
being  used  for  the  mean  daily  motion. 


CELESTIAL   MECHANICS:   LEUSCHNER  11 

(1)  CERES. 

The  first  and  largest  of  the  minor  planets  was  discovered  1801, 
January  1,  by  Piazzi  in  Palermo.1 

Piazzi  assumed  that  the  object  was  a  comet,  but  several  astronomers 
succeeded  in  proving  from  the  22  meridian  observations  near  the 
stationary  point  over  an  heliocentric  arc  of  9°  that  it  was  a  planet 
moving  in  a  nearly  circular  orbit;  thus  Burckhardt2  computed 
Elements  A,  Olbers3  the  circular  Elements  B,  Piazzi*  the  circular 
Elements  C.  Only  the  computation  by  Gauss,5  Elements  D,  was 
accurate  enough,  especially  in  the  determination  of  perihelion  and 
eccentricity,  to  indicate  where  the  planet  might  be  found  the  following 
year. 

Olbers  found  Ceres  again  1802,  January  1,  %°  from  the  predicted 
place,  near  the  place  where,  three  months  later,  he  discovered  the 
second  of  the  minor  planets.  The  new  observations  naturally 
increased  the  accuracy  of  the  elements  notably;  thus  Gauss6  computed 
Elements  E,  from  observations  in  1801,  and  January  1802;  represen- 
tation in  February  1802,  +7"  in  a,  —20"  in  8.  Burckhardt7  including 
the  perturbations  larger  than  30'  found  Elements  F. 

For  some  years  the  orbit  of  Ceres  was  investigated  by  Oriani, 
Burckhardt,  and  Gauss  by  taking  the  perturbations  into  account,  but 
the  efforts  of  Gauss  went  farther  than  those  of  the  others.  Burck- 
hardt8 started  with  the  computation  of  perturbations  at  intervals  of  two 
days,  and  later  computed  tables  founded  upon  them.  Oriani9  used 
Laplace's  method,  with  which  also  Gauss  started.  Gauss  developed 
the  perturbations  first  in  1802,  together  with  Elements  VIII,  G,  and 
formed  tables  of  perturbations10  and  later  in  180511  when  he  used 
the  same  interpolatory  development  of  the  perturbative  function  as 
Hansen  later  used  in  1830. 

The  orbit  computation  was  taken  up  later  by  Heiligenstein.12  He 
derived  Elements  H  from  the  oppositions  1818,  1820,  1821,  1822,  1825, 
1826,  1827,  with  special  perturbations  of  the  elements  by  Jupiter, 
(mass  1/1053.924).  Representation  of  the  normal  places  — 10"  to 
+6"  in  mean  longitude.  Correction  to  the  ephemeris  for  1830  April, 
May,  —6"  in  a,  —10"  in  8. 

Heiligenstein's  ephemeris  deviates  15'  from  the  ephemeris  in  B.  J. 
1830,  which  was  based  on  the  elements  of  Gauss  (XIII,  1809),  using 
the  tables  of  perturbations  by  Gauss  and  an  empirical  correction  by 
Encke  of  14'  to  the  mean  longitude  determined  from  the  last 
observations.18 


12  CELESTIAL   MECHANICS:   LEVSCHNER 

In  B.  J.  for  1831  Encke14  gives  an  ephemeris  from  new  Elements 
I  of  his  own  based  on  the  oppositions  1820,  1821,  1822,  1825. 
Jupiter  mass  1/1053.924.  Special  perturbations  by  Jupiter  only. 
Representation: 

1820      1821        1822       1825       1827       1829 
in  a  —  6"       +2"      —4"      —3"      —2"      —27" 
in  8      0"          0"       +6"       +1"          0"          11" 

In  B.  J.  1832  to  1836  the  ephemerides  by  Heiligenstein  were  published. 
Later  the  computation  by  Encke  and  Wolfers  was  used  to  1871. 

In  the  meantime  Damoiseau16  had  given  expressions  for  the  per- 
turbations containing  a  large  number  of  terms  but  the  individual 
coefficients  do  not  seem  to  be  very  exact,  according  to  Hill. 

For  the  use  of  the  American  Ephemeris,  E.  Schubert16  undertook 
to  correct  the  elements  by  250  observations  in  14  oppositions,  1832- 
1854,  using  special  perturbations  of  the  elements  by  Jupiter  as  com- 
puted by  Encke  and  Wolfers  but  corrected  for  the  secular  variation 
of  the  obliquity.17  Elements  J.  Residuals  in  a  — 22"  to  +21",  in 
8  —8"  to  +8" ;  </>  corrected  according  to  A.  J.,  Vol.  5,  p.  73.  A  further 
correction  of  the  elements  by  Schubert18  was  based  on  only  four 
normal  places  in  1853,  1854,  1855,  1857;  he  applied  the  special  per- 
turbations of  the  elements  by  Jupiter  and  Saturn.  Representation  of 
the  normals  ±0",  "by  which  the  correctness  of  the  whole  is  proved." 
Elements  K. 

Godward19  repeats  the  process  of  Heiligenstein,  Encke,  Wolfers, 
Schubert.  The  errors  for  fifteen  oppositions  1857  to  1876  of  the 
ephemerides  in  Nautical  Almanac  which  include  the  perturbations 
of  Venus,  the  Earth,  Mars,  Jupiter,  Saturn  gave  by  a  least  squares 
solution  the  Elements  L.  Ephemerides  by  these  elements  were  given 
in  the  Nautical  Almanac  to  1913. 

The  corrections  to  Encke's  ephemerides  increased  after  28  years 
to  ±38  in  a  ±20"  in  8. 

The  corrections  to  Schubert's  ephemerides  increased  after  23  years 
to  +68  in  a  ±40"  in  8. 

The  corrections  to  Godward's  ephemerides  increased  after  36  years 
to  +2a  in  a  ±10"  in  8. 

For  the  purpose  of  illustrating  his  modified  form  of  computing 
absolute  perturbations  Hill20  computed  the  first  order  perturbations 
of  Ceres  by  Jupiter  starting  with  the  first  elements  by  Schubert  (un- 
corrected).  It  was  found  that  the  osculating  mean  motion  differed 


CELESTIAL   MECHANICS:    LEUSCHNER  13 

widely  from  the  mean  mean  motion.  An  arbitrary  value  was  substi- 
tuted. The  Jupiter  mass  is  taken  to  be  1/1047.355.  The  expressions 
for  the  periodic  terms  of  the  perturbations  are  given.  In  order  to 
arrive  at  mean  elements  as  well  as  to  see  how  closely  the  perturba- 
tions represent  the  observations,  ten  normal  places — 1802,  1807,  1830, 
1857,  1863,  1866,  1873,  1883,  1885,  1890,  were  formed.  Secular  per- 
turbations of  Mars,  Jupiter,  and  Saturn  were  computed  by  the  method 
of  Gauss.  The  periodic  perturbations  by  Mars  and  Saturn  were  taken 
from  the  tables  of  Damoiseau.  Preliminary  elements  and  a  least 
squares  solution  led  to  mean  Elements  M.  The  residuals  are  — 40" 
to  +40"  in  hel.  longitude,  —20"  to  +13"  in  geoc.  latitude.  Hill 
originally  intended  to  enlarge  and  complete  his  theory  of  Ceres;  for 
this  purpose  he  collected  the  observations  into  75  normals  from  1801- 
1897.21  He  published  the  positions  because  he  did  not  expect  to  finish 
the  work.  The  collection  is  not  complete. 

As  an  extension  of  Hill's  work  Merfield22  has  given  a  computation 
of  the  secular  perturbations  of  Ceres  arising  from  the  action  of  the 
eight  Major  Planets.  From  Hill's  theory  and  his  mean  elements 
using  the  method  of  Gauss  as  set  forth  by  Hill  the  numerical  values 
of  the  action  of  the  planets  were  derived. 

M.  Wolf23  has  developed  the  expression  (p)  according  to  the  theory 
of  Gylden  in  the  case  of  Ceres.  Cf.  Tisserand,  Mecanique  Celeste, 
Vol.  4. 

M.  Viljev24  has  published  tables  of  absolute  perturbations  of  Ceres 
after  the  method  of  Hansen. 

REFERENCES 

'BODE'S   B.  J.    1804,   p.  249.    B.  J.  "B.  J.  1831,  p.  277.    A.  N.  vol.  27, 

1805,    p.    202.      v.    Zach,    Monatliche  p.  177. 

Correspondance.     Not   available  here.  1<s  Connaissance  des  Temps  1846.    Ad- 

Lalande,  Connaissance  des  Temps  de  ditions  p.  32.    Not  available  here. 

1'annee  xiii.    Witt,  H.u.  E.  vol.  14.  "A.  J.  vol.  3,  p.  153;  p.  162. 

BoDEsB.  J.    1804,  p.  255.  "Nautical  Almanac  1837.    B.  J.  1838, 

8  BODE  s  B.  J.    1804,  p.  256.  D  ose 

CODE'S  B.  J.    1804,  p.  259.  «A'  T       i   ,       „ 

CODE'S   B.   J.    1805,   p.   94.    Gauss.  »M  N  vol  38  D   119 

Werke  Bd.  6,  p.  200.  *>  f '  T  Vo1    16   n  '  87 

•  GAUSS.    Werke  Bd.  6,  p.  207.  »  J  J;  *£  3/51 

7  BODE'S  B.  J.    1805,  p.  96.  M  fjf  V  ™l  R7P'    *ai 

8  A  J  vol  16  p  57  '  N<  vo  '  67'  p>  551< 

9v.'   Zach'.      Monatliche    Correspon-  "WOLF.      Bur    les    termes    elemen- 

dance  1802,  Dec.    Not  available  here.  taires    dans    I  expression     du     rayon- 

10  GAUSS.    Werke  Bd.  7,  p.  375.  vecteur.    Stockholm,  1890. 

11  GAUSS.    Werke  Bd.  7,  p.  401.  Publications  de  1'Observatoire  Cen- 
13  A.  N.  vol.  7,  p.  413.  tral  Nicolas,  Poulkovo.    Not  available 
18  B.  J.  1830,  p.  245.  here. 


14 


CELESTIAL   MECHANICS:    LEUSCHNER 
TABLE  I.— Elements— (1)  Ceres 


Letter 

Date 

MT 

L 

IT 

Q 

i 

o       /           » 

O           t                • 

o      t             * 

e      /     ^^ 

A.  .  .  . 

1801  Jan.  1  .  3328 

68  59  37 

248  59  37 

80  58  30 

10  47 

B  

1801  Jan.  1  

68  35  51.5 

80  22  45 

11     3  36 

C.  .  .  . 

1801 

68  46  41 

80  46  48 

10  51  12 

D.  .  . 

1801 

Palermo  .... 

76  28  14 

150  33  20 

81     2  35 

10  36  30 

E  

1801  

Palermo  .... 

77  27  31 

145  57  15 

80  58  40 

10  37  57 

F  

1802  

155  32  35 

146  44  37 

81     5  35 

10  36  52 

G  

1801  

Seeberg  

77  19  34.9 

146  33  37 

80  54  59 

10  37  56.0 

H.... 

1818  Oct.  15.0.. 

G6ttingen.  . 

28  21  52.291 

148  2  14.084 

80  48  32.192 

10  38  21.682 

I  

1822  Jan.  22.0.  . 

Gottingen.  . 

127  36  44.2 

147  36  57.6 

80  41  55.0 

10  38     7.7 

M 

o       /           • 

J  

1854  Jan.  0  

Washington. 

113  22  25.08 

148  55  23.41 

80  50  50.79 

10  37     8.54 

K.... 

1854  Jan.  0  

Washington. 

113  18  22.40 

148  56     3.72 

80  50  31.11 

10  37     4.76 

L  

1854  Jan.  0  

Washington  . 

113  22  11.73 

148  55  26.54 

80  50  31.06 

10  37     5.81 

L 

o       /           • 

M.... 

1850  Jan.  0.0.  .  . 

Greenwich.. 

309  30  32.4 

148  28  32.5 

80  48     5.6 

10  37     6.2 

Letter 

Date" 

MT 

r 

M 

Equinox 

Author 

1801  Jan  1  3328 

0       /                » 

2     5 

* 

859  05 

Burckhardt  I 

B 

1801  Jan   1 

0 

786  528 

Olbers 

c 

1801 

0 

795  937 

Piazzi 

D  

1801  

Palermo  .... 

4     2  45 

784.254 

Gauss  I 

E  
F  . 

1801  
1802     

Palermo..  .  . 

4  40  10 
4  31  25 

769.7925 
771.363 



Gauss  VII 
Burckhardt  II 

G.  ... 

1801   

Seeberg  .  . 

4  31  17.8 

770.7951 

Gauss  VIII 

H  
I  

J  
K.... 
L  

M.... 

1818  Oct.  15.0.. 
1822  Jan.  22.0.. 

1854  Jan.  0  
1854  Jan.  0  
1854  Jan.  0  

1850  Jan.  0.0... 

Gottingen.  . 
Gottingen.  . 

Washington. 
Washington. 
Washington. 

Greenwich  .  . 

4  31     5.183 
4  .31  18.0 

4  24  28.41 
4  24  29.36 
4  24  29.65 

4  29  57.8 

771.2273825 
770.72468 

769.63875 
769.62476 
769.64746 

770.718276 

1818.00 
1810.00 

1854.00 
1854.00 

1850.00 

Heiligenstein 
Encke 

Schubert 
Schubert 
Godward 

Hill* 

*Mean  elements. 


CELESTIAL   MECHANICS:    LEUSCHNER  15 

(2)  PALLAS 

Discovered  by  Gibers  at  Bremen  1802,  March  28.1  Gibers  attempted 
to  compute  a  circular  and  a  parabolic  orbit  for  the  new  planet,  both 
of  which  failed.  His  computation  showed  the  orbit  had  a  large 
inclination  and  considerable  eccentricity. 

From  observations  extending  from  April  1  to  July  8,  Gauss2  com- 
puted Elements  A  (Gauss  V).  They  are  improvements  on  preliminary 
sets.  With  these  elements  an  ephemeris  for  1803  was  computed. 

From  observations  extending  from  April  4  to  May  20,  Burkhardt3 
computed  Elements  B.  With  these  elements  Burkhardt  computed 
the  perturbations  in  longitude,  latitude,  and  radius  vector  covering 
the  period  April  4  to  May  20. 

The  planet  was  reobserved  by  Harding  1803,  Feb.  21st.  The  com- 
parison between  Gauss'  ephemeris  and  observations  was  as  follows: 

1803  Aa  AS 

Feb.  21  +2'  02"  —34" 

Feb.  23  +2    35  —57 

On  the  basis  of  these  residuals,  Gauss4  improved  Elements  A 
Gauss  (V).  These  new  Elements  C  (Gauss  VI)  represent  the  obser- 
vations as  follows: 

1803  Aa  AS 

Feb.  21  -  20"0  +  15"8 

Feb.  23  +  7.8  —  7.7 

From  a  set  of  elements,  based  on  oppositions  1804,  1805,  1807,  1808, 
Gauss5  derived  an  improved  set  of  Elements  D  from  a  least  squares 
solution.  This  solution  includes  also  the  oppositions  1803  and  1809 
and  forms  the  basis  for  the  computation  of  perturbations  as  outlined 
below. 

Gauss6  first  attempted  to  construct  tables  of  perturbations  for  the 
four  known  minor  planets,  but  the  large  eccentricity  and  inclination 
forced  him  to  formulate  a  theory  for  Pallas  based  on  the  variation 
of  the  elements  expressed  analytically  and  integrated  by  mechanical 
integration.  From  two  successive  calculations  of  the  special  pertur- 
bations, due  to  Jupiter,  Gauss  derived  the  improved  Elements  E, 
which  represented  the  heliocentric  longitudes  for  the  first  seven  oppo- 
sitions within  ±8". 

In  1811  Gauss6  began  his  first  computation  of  general  perturbations 
due  to  Jupiter.  For  this  purpose  he  used  Laplace's  elements  of  Jupiter, 


16  CELESTIAL   MECHANICS:   LEUSCHNER 

epoch  1805,  and  his  own  elements  of  Pallas  for  the  same  epoch.  This 
computation  led  to  a  set  of  mean  Elements  (F).  With  these  mean 
elements  for  epoch  1810  and  similar  elements  for  Jupiter  (Laplace) 
a  second  computation  of  general  perturbations  due  to  Jupiter  was 
undertaken.  This  computation  led  to  the  following  results: 

The  mean  motion  of  Pallas  oscillates  between  18/7  of  QJ.  motion 
±0".2153,  and  1894  revolutions  of  Pallas  =  737  of  Jupiter.  A  new 
value  for  Jupiter's  mass  =  1/1042.86.  Then  follows  (1816-1817)  the 
computation  of  perturbation  tables  due  to  Jupiter,  Saturn  and  Mars. 
In  this  latter  work,  Gauss  was  assisted  by  Encke  and  Nicolai. 

In  Astronomisches  Jahrbuch  1816,  page  234,  Bode  gives  the  best 
set  of  elements  by  Gauss  up  to  that  time  (Elements  G). 

About  1824,  Encke7  used  Gauss'  elements  based  on  early  oppositions 
and  computed  the  perturbations  due  to  Jupiter.  He  reports  that 
Gauss'  elements  with  Jupiter's  perturbations  represent  the  opposition 
of  1823  as  follows: 

1823  Aa  AS 

Oct.  9  +13:2  +25^6 

He  then  gives  Elements  (H)  for  the  epoch  1826,  and  with  these 
computes  the  next  ephemeris. 

For  the  opposition  in  1825,  Encke8  reports  that  the  correction  to 
the  ephemeris  is  very  large.  But  if  the  perturbations  are  included, 
the  difference  between  observation  and  computation  is  as  follows: 

1825  Aa  AS 

March  23  +4276  —  33"2 

He  then  gives  a  set  of  elements  for  the  epoch  1827,  and  computes 
an  ephemeris  for  1827. 

By  1834  Encke9  reports  a  deviation  of  Pallas  from  computed 
places  amounting  to  5'.  He  states  this  may  be  due  to  use  of  Laplace's 
value  for  Jupiter's  mass.  It  will  be  necessary  to  recompute  elements 
covering  all  observations. 

In  A.  N.  No.  636,  Encke  publishes  osculating  Elements  I  for  each 
year  from  1831  to  1838;  his  fundamental  starting  elements  are  for 
the  epoch  of  1810,  January  0.  In  B.  J.  1838,  p.  286,  Encke  draws  atten- 
tion to  an  error  which  he  had  committed  in  neglecting  the  corrections 
for  the  secular  variation  of  the  obliquity.  In  the  British  Nautical 
Almanac  for  1837  Airy  points  out  this  error  to  which  Encke  refers. 

Galle10  undertook  the  reinvestigation  of  the  orbit  based  on  opposi- 
tions 1816, 1821, 1827,  1830,  1834,  1836,  making  use  of  Airy's  value  for 
the  mass  of  Jupiter  1/1048.69.  The  former  perturbations  were  retained 


CELESTIAL   MECHANICS:   LEUSCHNER  17 

except  for  the  change  due  to  Jupiter's  mass.    The  resulting  Elements 
J  represent  the  heliocentric  longitude  and  latitude  as  follows : 

1816  1821  1827  1830  1834  1836 

AL  —19"          +34"          +4"          —14"          +5"         —13" 

AB  —  1  +4  +6  —10  +1  +4 

Galle  states  these  differences  may  be  accounted  for  if  the  perturba- 
tions of  Saturn  and  Mars  were  taken  into  account. 

In  A.  N.  No.  636  osculating  Elements  K  are  published  for  each 
year  from  1839  to  1850.  These  were  computed  by  Galle.  The  start- 
ing elements  are  those  for  epoch  1810,  January  0.  In  computing  the 
special  perturbations,  Encke  and  Galle  used  mass  of  Jupiter  1/1053.924. 

From  1851  to  1870  Galle11  continues  the  special  perturbations  by 
Jupiter  and  later  with  the  elements  of  Giinther  also  those  by  Saturn. 
These  were  used  in  computing  the  ephemerides  published  in  the 
Astronomiches  Jahrbuch  from  1862  to  1870.  (See  Elements  L.) 

Beginning  with  the  year  1871  and  continuing  to  1919,  the  Jahrbuch 
published  and  used  Farley's12  osculating  elements  for  computing  the 
ephemeris.  (See  Elements  M  and  N).  Farley's  computation  includes 
the  perturbations  by  Venus,  Earth,  Mars,  Jupiter  and  Saturn.  His 
computations  are  also  the  basis  for  the  ephemerides  published  in  the 
British  Nautical  Almanac.  With  Farley's  elements  we  have  the  fol- 
lowing comparisons: 

Corrections  to  Ephemerides. 

1883            1892            1895            1906  1908  1914 

Aa           -1B4            -1?2            -1?0            -2?5  -5?4  -2s! 

A5           -J-2'7           +07           +0'8           +8'2  -14-0  +4'3 

In  Annales  de  1'Observatoire  de  Paris,  Vol.  I,  Le  Verrier  pub- 
lishes the  results  of  his  investigation  on  "Developpement  de  la 
fonction  perturbatrice  relative  a  Faction  de  Jupiter  sur  Pallas.  Calcul 
du  terme  dont  depend  une  inegalite  a  longue  periode  du  mouvement 
de  cette  derniere  planete."  Le  Verrier  states  that  the  aphelion  of 
Pallas  is  54°  from  the  intersection  of  the  orbit  with  Jupiter.  Conse- 
niiently  when  Pallas  is  at  aphelion  the  distance  from  Jupiter  is  in- 
creased on  account  of  the  great  inclination  of  the  orbit.  This  large 
inclination  diminishes  the  effect  due  to  the  large  eccentricity.  Le 
Verrier  gives  the  series  for  the  reciprocal  of  the  distance  in  a  more 
convergent  form  and  develops  the  equation  in  longitude  depending 
on  the  argument  18  QJ.  —  7  Pallas.  The  maximum  of  the  term  is  895". 
In  his  report  before  the  Paris  Academy,13  Cauchy  compares  his  theory 


18  CELESTIAL   MECHANICS:   LEVSCHNER 

with  the  results  by  Le  Verrier;  his  value  for  the  inequality  is  906". 
Cauchy's  investigation  is  more  fully  elaborated  by  M.  Puiseux  in  An- 
nales  de  1'Observatoire  de  Paris,  Vol.  VII.  In  Vol.  VIII,  ibid.,  Hoiiel 
has  recomputed  the  inequality. 

The  development  of  the  reciprocal  of  the  distance  was  later  extended 
by  Tisserand.14  He  shows  that  the  development  depending  upon  the 
inclination  and  eccentricity  is  divergent  in  some  parts  of  the  orbit 
of  Pallas  and  proceeds  to  give  the  analytical  development  and  to 
apply  it  to  the  case  of  Pallas. 

In  Bulletin  Astronomique  Vol.  XII,  1895,  M.  P.  Bruck  has  pub- 
lished the  results  of  his  work  on  'The  secular  variations  of  the  ellip- 
tic elements  of  Pallas  due  to  the  action  of  Jupiter."  He  used  the 
method  developed  by  Gauss  and  extended  by  Hill  and  Callandreau. 
He  utilizes  elements  by  Farley  for  the  epoch  1878. 

In  1910  Georg^  Struve15  published  his  results  on  "Die  Darstellung 
der  Pallasbahn  durch  die  Gauss'sche  Theorie  fur  den  Zeitraum  1803 
bis  1910."  The  result  of  his  work  based  on  63  normal  places  is  a 
more  accurate  value  for  the  mean  motion  of  Pallas  (769".1385). 
The  new  value  for  the  annual  motion  of  Pallas  compared  with  Jupiter 
becomes  18n'  —  7n  =  123".  The  deviation  between  observation  and 
computation  still  amounts  to  ±4'  which  is  attributed  to  the  second 
order  perturbations.  These  residuals  are  somewhat  reduced  by  empiri- 
cal terms. 

In  A.  N.  No.  205,  p.  225,  M.  Viljev  has  published  his  "Recherches 
sur  le  mouvement  de  Pallas."  He  attempts  to  reduce  the  residuals 
from  Struve's  work  (±4')  by  taking  into  account  second  order  terms 
in  the  general  perturbations,  employing  the  method  by  Hill.  He 
reports  his  results  as  negative. 

REFERENCES 

CODE'S  A.  J.  1805,  p.  102.  "British  Nautical  Almanac  1860  (not 

"BODE'S  A.  J.  1805,  pp.  106,  111,  228.  available  here). 

"BoDE's  A.  J.  1805,  pp.  181,  182.  13Tisserand  Merhaninup  TpWp  vol 

CODE'S  A.  J.  1806,  pp.  179-180.  ^c          Mecnam<lue  Celeste,  vol. 

'GAUSSWERKE.      VOl.  Vi,  pp.  3-24.  IV,  P.  ^/S. 

6  GAUSS  WERKE.    vol.  vii,  pp.  413-610.  Annales  de  1'Observatoire  de  Paris. 

'BoDE'sA.  J.    1828,  pp.  154,  157.  vol.  xv. 

"BoDE'sA.  J.    1829,  p.  158.  "Dissertation  Berlin.    Ebernig,  1910 

•  B.  J.  1837,  pp.  24&-250.  (not  available  here)  . 

*  PP' 


14329 

AB'.  J.  i851,  pp.  547  and  549.    A.  N.          A-  J-  B->  1910>  P-  188- 
vol.  55,  p.  194.  B.  A.  28,  p.  184. 


CELESTIAL   MECHANICS:    LEUSCHNER 


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20  CELESTIAL   MECHANICS:   LEUSCHNER 

(3)    JUNO 

Juno  was  discovered  by  Harding  at  Lilienthal  near  Bremen,  Sep- 
tember 1,  1804.  Gauss  computed  several  orbits  successively  correct- 
ing the  elements  by  new  observations.  Elements  VII1  corrected  with 
Bessel's  observations,  1807.  Ephemeris  for  1808,  April-December, 
approximately  given.  (Elements  A.)  A  number  of  additional  orbits 
were  computed  by  Gauss'  students  at  Gottingen  (Wachter,  Mobius, 
etc.). 

Wachter2:  Elements,  (eccentricity  omitted)  from  the  last  four 
oppositions  after  Gauss'  Method,  (Neue  Comment,  der  Gottingen  K. 
Societat,  Bd.  1)  including  the  opposition  1812.  Eccentricity  supplied 
from  Bode's  Astronomische  Jahrbuch,  1816,  p.  233.  (Elements  B.) 

Mobius3:  Oppositions  used:  1810,  1811,  1812,  1813.  Correcting 
mean  longitude  by  +4'55",  the  representation  of  the  observations 
1815,  March,  is  +8"  in  longitude,  and  —51"  in  latitude.  (Ele- 
ments C.) 

Nicolai4  at  Seeberg,  near  Gotha,  compared  Gauss'  observations 
with  the  orbit  of  Mobius,  (empirically  correcting  L) ,  determined  the 
oppositions  and  derived  new  elements.  Oppositions  used:  1811,  1812, 
1813,  1815.  "Juno  is  nearly  in  conjunction  with  Jupiter  and  the 
perturbations  may  be  large."  (Elements  D.) 

Taking  up  the  determination  of  the  large  perturbations  by  Jupiter 
by  the  method  of  special  perturbations,  Nicolai5  derived  a  new  set 
of  elements.  Oppositions  used:  1811,  1812,  1813,  1815,  1816,  1817, 
1818.  Representation  of  observations  in  1819,  Aa+2',6  AS  — 0'.2. 
(Elements  E.) 

Not  satisfied  with  the  representation  of  the  observations  by  his 
last  set  of  elements,  Nicolai6  extended  the  computation  and  deter- 
mined new  elements,  -which  represented  the  observations  of  the 
"Atom"  well  in  1820.  Oppositions  used:  1805-1819.  Special  per- 
turbations by  Jupiter.  Representation  of  the  observations  1820,  May, 
Aa  —7",  AS  —2".  (Elements  F.)  This  computation  of  the  special 
perturbations  was  continued  for  some  years. 

In  1823,  Nicolai7  derived  his  final  set  of  elements,  including  the 
determination  of  the  Jupiter  mass,  for  which  he  found  1/1053.924,  in 
agreement  with  the  value  Gauss  had  found  from  his  theory  of  the 
motion  of  Pallas.  The  representation  of  the  observations  cannot  be 
improved  by  taking  Saturn  or  Mars  into  consideration,  but  Nicolai 
considers  the  possibility  of  the  active  mass  of  Jupiter  changing  with 
the  body  acted  upon.  From  these  elements  osculating  elements  for 
1826  were  computed,  taking  account  of  the  special  perturbations  by 


CELESTIAL   MECHANICS:   LEUSCHNER  21 

Jupiter.  Berliner  Jahrbuch  uses  the  elements  by  Nicolai  until  1830. 
Fifteen  oppositions  used:  1804-1823.  Special  perturbations  by  Jupiter, 
(Saturn,  Mars,  negligible).  Residuals  in  longitude  — 23"  to  +27", 
still  show  a  run  with  the  period  of  Jupiter.  (Elements  G.) 

In  1832  new  elements  by  Encke8  were  introduced,  and  are  carried 
forward  with  special  perturbations  to  1865  by  Bremiker  and  Powalky, 
for  the  ephemerides  published  1832-1865.  Perturbations  by  Jupiter 
with  mass,  1/1053.924.  (Elements  H.) 

Damoiseau9  has  published  general  perturbations  in  the  Connais- 
sance  des  Temps. 

Hind10  took  over  the  work  started  by  Nicolai,  Encke,  and  Bremiker, 
to  compute  osculating  elements  for  each  opposition  by  special  per- 
turbations. The  ephemerides  are  published  in  the  Nautical  Almanac, 
and  the  Berliner  Jahrbuch.  As  a  basis  for  this  work  he  derived  new 
elements.  (Elements  I.) 

An  attempt  to  apply  Hansen's  method  of  determination  of  the 
general  perturbations  was  made  by  Berkiewicz,11  starting  with  Hind's 
elements.  The  perturbations  of  the  first  order  with"  regard  to  Jupiter, 
Mars,  and  Saturn  were  determined,  also  the  constants  of  integration 
leading  to  a  mean  motion,  814".090.  No  comparison  with  the  obser- 
vations is  attempted. 

Being  aware  that  the  corrections  to  the  ephemerides  computed 
according  to  Hind  had  increased  to  3'  in  1887,  Downing12  undertook 
to  correct  Hind's  elements.  The  errors  of  the  tabular  heliocentric 
places  published  in  Greenwich  Observations,  1864-1887,  are  discussed. 
Equations  for  the  longitude  and  latitude  corrections  were  set  up  ex- 
pressed in  terms  of  corrections  to  the  elements  and  combined  to  elimi- 
nate the  corrections  to  the  radius  vector.  The  mean  motion  is  included 
in  the  solution  and  receives  by  far  the  greatest  weight.  The  represen- 
tation of  the  oppositions  show  a  pronounced  run  in  Aa.  These  ele- 
ments were  used  for  the  computation  of  the  annual  ephemerides  to 
1913  in  the  Nautical  Almanac  and  to  1917  in  the  Berliner  Jahrbuch. 
(Elements  J).  Oppositions  used:  1864-1887.  Special  perturbations 
by  Venus,  the  Earth,  Mars,  Jupiter,  Saturn.  Representation  varies 
from  —3"  to  +4"  in  Aa  cos  8,  and  — 1"  to  +2"  in  AS.  Large 
residual  (—11")  for  1874.  Representation  for  1890  is  then  in  a+3", 
in  SihO"  against  — 65"  and  — 6",  according  to  Hind's  computations. 

Since  1917  Ephemeriden  der  Kleinen  Planeten  gives  mean  elements 
by  Boda13  derived  by  the  method  of  Brendel.  (Elements  K.)  Mean 
elements.  Perturbations  by  Jupiter  according  to  Brendel,  A.  N.,  Vol. 
195,  p.  417.  Expected  representation  ±0°.5  to  year  2000.  Opposi- 
tions not  stated. 


22 


CELESTIAL   MECHANICS:    LEUSCHNER 


The  absolute  perturbations  according  to  the  method  of  Hansen  (?), 
have  been  computed  by  Viljev.14 


REFERENCES 


'BODE'S    B.    J. 

Werke  Bd.  6. 
2  BODE'S  B.  J. 
8  BODE'S  B.  J. 
4  BODE'S  B.  J. 
6  BODE'S  B.  J. 
6  BODE'S  B.  J. 


1811,  p.  136.     Gauss 

1815,  p.  248. 

1817,  p.  213. 

1818,  p.  264. 
1820,  p.  200. 
1822,  p.  218. 


7  BODE'S  B.  J.    1826,  p.  224. 

8  A.  N.  Bd.  27,  p.  177. 


9  Connaissance  des  Temps,  1846.    Ad- 
ditions.   Not  available  here. 

10  Nautical  Almanac,  1859.    Appendix. 
Not  available  here. 

"A.  N.  vol.  72,  p.  1,  p.  145,  p.  289. 
"Nj.  M.  vol.  50,  p.  487. 
18  A. "N'r vol.  200,  p.  1. 
14  Bulletin-Soc.  Astr.  Russ.     vol.  22. 
Not  available  here. 


TABLE  3.— Elements— (3)  Juno 


Letter 

Epoch 

M.  T. 

L+rM. 

1C 

Q 

i 

A  
B 

1805  
1811 

Gottingen  li 
Gottinpen     .      ././ 

42  37     3.7 
177  48  21.0 
95  29  53.2 
230  11  34.2 
117  45     2.84 
230     9  22.03 
95  25    9.82 
351  43  27.3 
58  34     1.0 
58  34     1.83 
324  12 

53  19     0.2 
53  15  10.1 
53    6  43.0 
53  14  53.8 
53  32  56.09 
53  31     6.52 
52  58  35.89 
53  11  13.4 
54     9     3.3 
54     9     3.82 
55  36 

171     4  28.2 
171     9  16.7 
171     6  45.0 
171     9  58.9 
i71     6  50.23 
171     8  11.08 
171     6  28.52 
170  56  57.4 
170  59  49.7 
170  5\45.69 
170  42 

13  4  26.2 
13  4  17.2 
13  4  12.9 
13  4     0.1 
13  3  37.29 
13  3  47.20 
13  4  18.99 
13  3  28.4 
13  2  58.8 
13  2  58.45 
13  2 

C  .. 

1810  

Gottingen  L 
Gottingen  L* 
Mannhein*  ....// 
Mannheim  Lt 
Gottingen  L 
Berlin  ft 

D  
E  
F  
G  
H  
I  
J  
K  

1815  Dec.  31.0..'.. 
1819  

1820  May  11  
1810  
1826  Nov.  1  
1861  Nov.  21.0..  . 
^61  Nov.  21.0.  .. 
1900  Jan  0 

Greenwich  .  .  .  .  t* 
Greenwich  .  .  .  .  L> 
It 

Letter 

Epoch 

M.  T. 

»• 

P 

Equinox 

Authority 

A  

1805  

Gottingen  

Of                f 

14  48  11.5 

813  8468 

GaussVII 

B  

c 

1811  
1810 

Gottingen  
Gottingen 

14  44  IX 
14  43     95 

813.25748 
812  7140 

1811 
1810 

Wachter 
Mobius 

D.... 
E 

1815  Dec.  31.0.. 
1819 

Gottingen  
Mannheim 

14  43  28.84 
14  53  17  44 

812.9304 
813  86981 

1816.0 
1819 

Nicolai 

F  
G.... 
H  

1820  May  11.... 
1810  
1826  Nov.  1.  .  .  . 

Mannheim  
Gottingen  
Berlin  

14  55     1.78 
14  44  39.19 
14  53  22  6 

814.40238 
813.4837354 
813  88514 

1820  May  11 
1810 
1810 

Nicolai 
Nicolai 
Encke 

I  

1861  Nov.  21.0.. 

Greenwich  

14  47  14.1 

813  34555 

1861  Nov  21  0 

Hind 

J.  ...  . 

1861  Nov.  21.0.. 

Greenwich  

14  47  13  81 

813  35271 

1861  Nov  21  0 

Downing 

K  

1900  Jan.  0  

14  50 

813.434 

1900 

Boda 

CELESTIAL   MECHANICS:   LEUSCHNER  23 

(4)  VESTA 

Vesta  was  discovered  by  Olbers1  at  Bremen  on  March  29,  1807. 

Preliminary  elements  were  computed  by  Gauss2  and  also  by  Burk- 
hardt.3  The  third  set  of  Elements  A  by  Gauss  is  based  on  obser- 
vations from  March  29  to  July  11.  They  were  used  to  compute  the 
ephemeris  for  1808-1809. 

The  preliminary  work  of  Gauss  was  continued  by  Gerling,4  who 
supplied  the  ephemeris  for  a  number  of  years.  His  last  set  of  Ele- 
ments B,  are  based  on  the  first  six  oppositions. 

Burkhardt's  preliminary  work  was  continued  by  Daussy5  (refer- 
ence not  available  here).  In  his  work  he  took  into  consideration  the 
perturbations  by  Jupiter,  Saturn,  and  Mars,  and  was  able  to  represent 
the  first  seven  oppositions  satisfactorily.  On  account  of  the  small 
eccentricity  and  inclination,  the  methods  of  La  Place  and  Le  Verrier 
were  sufficient. 

On  account  of  increased  error  in  the  ephemeris  for  1818  based 
on  Gerling's  first  elements,  Encke0  computes  a  set  of  Elements  C, 
based  on  oppositions  1812,  1815,  1816,  and  1818. 

In  Astronomisches  Jahrbuch  1829,  pp.  156-158,  Encke  gives  the 
results  of  his  work  on  Vesta  based  on  fourteen  oppositions  by  also 
using  Nieolai's  value  of  Jupiter's  mass  (1/1054),  for  the  perturbations 
due  to  Jupiter.  He  also  makes  use  of  Daussy's  tables  for  the  per- 
turbations of  Saturn  and  Mars.  His  new  set  of  Elements  D  repre- 
sents the  oppositions  from  1807  to  1825  within  ±6".  Elements  D 
are  then  brought  up  to  the  epoch  of  1827.  Astronomisches  Jahr- 
buch from  1830  to  1866,  contains  the  elements  and  ephemerides  com- 
puted by  Encke.  (See  Elements  E.)  The  method  of  computation7 
was  that  of  the  variation  of  the  elements  by  special  perturbations  of 
Jupiter.  In  this  work  Encke  was  assisted  by  Bruhns  and  Schiaparelli. 
In  A.  N.  332,  Encke  comments  on  the  poor  results  obtained  for  planets 
(1),  (2)  and  (3),  by  using  Laplace's  value  for  the  mass  of  Jupiter 
and  also  publishes  a  new  value  of  Jupiter's  mass  (1/1050),  obtained 
from  the  results  of  Vesta. 

The  Berliner  Jahrbuch  for  1868  publishes  mean  Elements  F  by 
Briinnow.8  They  are  also  published  in  Watson's  Theoretical  Astron- 
omy. Briinnow  completes  the  work  of  Wolfers  and  Galle9  who  de- 
veloped expressions  for  the  perturbations  in  longitude  and  radius 
vector  after  Hansen's  method.  Bninnow's  elements  represent  the 
oppositions  from  1810  to  1851  within  —6"  to  +10".  For  Jupiter's 
mass  he  used  1/1050. 


24  CELESTIAL  MECHANICS:   LEVSCHNER 

From  1871  to  1910  the  Berliner  Jahrbuch  publishes  elements  (see 
Elements  G),  by  Farley.10  His  work  is  based  on  twelve  oppositions, 
1840  to  1855.  This  work  forms  the  basis  for  later  investigations  of 
the  general  perturbations. 

Probably  the  most  extensive  work  on  a  minor  planet  are'  the  tables 
of  Vesta  by  Leveau  published  in  Annales  de  FObservatoire  de  Paris 
Memoires,  XV,  XVII,  XX,  XXII,  XXV.  The  method  applied  by 
Leveau  is  that  of  Hansen  "Auseinandersetzung  einer  Zweckmassigen 
Methode  zur  Berechnung  der  Absoluten  Storungen  der  Kleinen 
Planeten,  I,  II,  III."  The  explanation  for  the  choice  of  this  method 
is  that  the  application  to  Vesta  is  a  preparatory  study  to  the  motion 
of  Pallas,  as  Gauss'  theory  of  Ceres  was  a  preliminary  study  to  his 
theory  of  Pallas.  Memoir  XV  contains  the  perturbations  of  the 
first  order  of  the  masses  of  Venus,  Earth,  Mars,  Jupiter,  Saturn, 
Uranus,  and  Neptune.  The  memoir  concludes  with  the  determina- 
tion of  the  constants  of  integration  and  the  expressions  for  nSz,  v, 
u  seci  and  a  representation  of  an  observation  1858,  April,  23.5  as 
follows:  Aa= — 0s.!  and  AS= — 0".4.  Memoir  XVII  contains  the 
terms  depending  upon  the  square  of  the  mass  of  Jupiter  in  n8z 
and  2i/.  The  effect  on  u  seci  becomes  noticeable  after  100  years. 
Memoir  XX  contains  the  terms  depending  upon  the  product  of  the 
masses  and  concludes  with  a  set  of  mean  elements  and  corresponding 
expressions  for  nSz,  v,  and  u  seci.  The  mean  elements  given  in  memoir 
XX  are  slightly  changed  in  memoir  XXII  (see  Elements  H),  on  ac- 
count of  some  perturbations  of  the  second  order  that  Farley  had  in- 
cluded and  which  form  the  basis  of  Leveau's  work.  Memoir  XXII 
contains  the  comparisons  with  215  normal  places  founded  on  5000 
observations  extending  from  1807  to  1889.  The  computation  has  been 
changed  to  conform  to  the  solar  tables  by  Newcomb.  The  correction 
to  the  mean  motion  is  zero.  The  new  Elements  I  represent  the  obser- 
vations in  right  ascension  between  — 2"  and  +4"  and  in  declination 
between  — V  and  +4".  A  new  value  of  Jupiter  1/1046,  and  Mars 
1/3648000.  The  perturbations  are  collected  in  tables  introducing  the 
mean  anomaly  and  corrective  terms  for  the  eccentric  anomaly.  Mem- 
oir XXV  contains  some  supplementary  terms  depending  upon  the 
product  of  the  masses.  One  of  the  larger  terms  has  a  period  of  3000 
years.  The  effect  of  the  critical  terms,  which  are  mentioned  in  the 
beginning  of  his  work,  shows  itself  by  comparing  these  terms  before 
and  after  integration.  The  coefficients  of  corresponding  terms  are 
ten  or  twenty  times  larger,  whereas  the  other  terms  mostly  decrease. 

Leveau  has  made  a  comparison  between  observations  and  calcu- 
lation of  later  positions11  showing  the  residuals  as  obtained  from 


CELESTIAL   MECHANICS:   LEUSCHNER  25 

the  British  Nautical  Almanac,  and  his  own  computations.  The  fol- 
lowing comparisons  are  illustrations  of  the  results: 

1890          1892          1894          1896  1898 

Nautical  Almanac    Aa  +ls.18     +18.00      +18.73      +ls.71  +2891 

AS  +0".9     —  5".8       +8".9      —  2".0  +  16".3 

Leveau                  '    Aa  +03.03     +0S.01      +0S.25      +0S.06  +03.19 
A8+0".4      +0".5       +2".0 


Further  results  of  Leveau's  theory  are  published  in  Comptes  Rendus, 
T.  145,  p.  903-906,  "Determination  des  Elements  Solaires  et  des 
Masses  de  Mars  et  de  Jupiter  par  les  Observations  Meridiennes  de 
Vesta."  Extending  the  comparison  with  the  meridian  observations 
from  1807  to  1904  and  taking  into  consideration  the  masses  of  Jupiter 
and  Mars  and  also  the  solar  elements,  Leveau  determines  a  new  set 
of  smaller  corrections  to  the  elements  of  Vesta  and  for  Jupiter's  mass, 
1/1046,  and  mass  of  Mars  1/3601280.  The  tables  of  residuals  shows 
the  poor  quality  of  the  meridian  observations  before  1826.  A  period 
of  36  years,  (three  revolutions  of  Jupiter,  or  ten  of  Vesta)  ,  points  to 
the  effect  of  the  critical  terms  in  the  residuals  ;  the  amplitude  is  about 
1".  The  effect  of  the  earlier  observations  on  the  value  for  the  mean 
motion  is  also  illustrated  by  the  last  residuals. 

In  Annales  de  1'Observatoire  Astronomique  de  Toulouse,  T.  I.,  B.  1 
to  B.  90,  M.  J.  Perrotin  published  his  extensive  investigation  on  the 
"Theorie  de  Vesta,"  applying  the  method  of  Le  Verrier  (Annales  de 
1'Obs.  de  Paris,  T.  X.).  The  method  consists  first  of  deriving  mean 
elements  from  previous  osculating  elements  by  computing  provisional 
periodic  perturbations  and  applying  these  to  the  osculating  elements 
for  a  first  approximation.  In  order  to  avoid  considerable  labor,  Per- 
rotin starts  with  a  certain  fixed  major  axis  and  develops  corrective 
terms  for  the  variation  in  the  assumed  value.  The  final  mean  motion 
is  determined  from  two  extreme  groups  of  observations  in  1807  and 
1876  when  the  planet  was  near  the  same  place  in  its  orbit.  The  per- 
turbative  function  is  developed  by  Le  Verrier  to  the  seventh  degree  in 
the  inclination  and  eccentricity  ;  thus  Perrotin  includes  terms  of  10n'— 
3n.  Venus,  Earth,  Mars,  Jupiter,  and  Saturn  are  taken  into  account. 
Derived  from  the  secular  terms  e  is  always  smaller  than  0.15,  the 
mean  motion  of  the  perihelion  is  +38",  that  of  the  node  —  38",  and 
the  inclination  remains  less  than  9°.  The  terms  of  the  second  order 
are  then  considered.  Those  due  to  the  square  of  Jupiter's  mass  of  the 
second  degree  are  small.  Those  depending  on  the  product  of  the 
masses  are  more  important,  especially  those  depending  upon  5n"  —  2n', 


26  CELESTIAL   MECHANICS:   LEVSCHNER 

— 2n"+4n' — n,  2n"+9n' — 3n.  For  getting  these  terms  the  develop- 
ment of  the  perturbative  function  is  used  to  the  seventh  degree  and  for 
determining  the  terms  of  the  eighth  degree  the  method  of  Cauchy,  as 
extended  by  Puiseux,  is  used.  No  comparison  with  observations  is 
attempted. 

REFE.RENCES 

1  BODE'S  B.  J.  1810,  p.  194,  6  Connaissance  des  Temps,  1818,  1819, 

'BODE'S  B.  J.    1810,  pp.  198,  213;  1812,      1820. 
p.253;  Gauss  Werke.    vol.  vi. 

3 BODE'S  B.  J.  1810,  p.  199;  Annales 
de  1'Observatoire  Astronomique  de 
Toulouse,  vol.i,  p.  B4.  10  British  N.  A.  1860. 

4B.  J.  1814,  p.  253;  B.  J.  1817,  p.  255;         "Bull.  Astr.    vol.  19,  p.  434;  Comptes 
B.  J.  1819,  p.  224.  Rendus,  T.  135,  p.  525. 


CELESTIAL   MECHANICS:    LEUSCHNER 


27 


3rd  set  of  preliminary 
Based  on  first  six  op 


Al. 
A57 


B.  J.  1831, 

B.  J.  1871. 
Mem.  XX 
Mem.  XX 


- 


OOCO<NtO 


^t^coasTjtiot^co® 

1000000000 


83ig2   :   :   :   :   : 

8S88   :   :   :   :   : 


r-*^ 

1-^r^^HCO^^CO|LCfcO 


«j  PQ  CJ  P  W  fe  O  W  HH' 


28  CELESTIAL  MECHANICS:   LEUSCHNER 

(10)  HYGIEA 

De  Gasparis  at  Naples  announced  to  the  General-Secretary  of  Cor- 
rispondenza  Scientifica  that  he  had  discovered  this  planet  April  12, 
1849.  The  magnitude  was  9-10.1 

A  large  number  of  preliminary  orbits  were  computed  by  Encke,2 
Luther,3  Brorsen,4  Quirling,5  d'Arrest,6  Hensel,7  Santini,8  from  observa- 
tions in  the  first  opposition  with  mean  motions  ranging  from  601"  to 
652".  The  first  more  certain,  Elements  A,  were  those  by  d'Arrest9 
from  observations  in  1849  and  1850.  These  elements  were  published 
in  B.  J.  1853-55,  brought  forward  with  special  perturbations  by  Jupiter. 

In  B.  J.  for  1856  two  sets  of  elements  were  published:  one  by  Cheval- 
lier  at  Durham,  B,  from  observations  in  1851  and  1852;  and  one,  C, 
by  Zech  at  Tubingen.  The  latter  (corrected  for  an  error10  which  did 
not  affect  the  ephemeris)  were  based  upon  four  oppositions  and  repre- 
sented the  opposition  1854  by  +0s-t>  in  a  and  — 1"  in  8.  This  was  the 
beginning  of  Zech's  important  work  which  made  this  planet  suitable 
for  testing  theories  in  the  case  of  near  commensurability. 

In  B,  J.  1858,  new  Elements  D,  by  Zech,11  are  published,  based  upon 
five  oppositions  taking  the  perturbations  by  Jupiter,  Saturn,  and  Mars 
into  account.  These  elements  are  very  closely  the  same  as  those  v. 
Zeipel  selected  from  Zech's  manuscript.  In  this  the  general  perturba- 
tions by  Jupiter,  Saturn,  and  Mars  were  computed  and  partly  tabu- 
lated. The  basic  elements  for  this  computation  were  founded  on 
eight  oppositions.  After  Zech's  death  in  1864  his  computations  of 
the  general  perturbations  of  the  planets  (5)  and  (10)  were  discon- 
tinued by  the  Recheninstitut,  whereas  his  elements  and  special  per- 
turbations for  (10)  Hygiea  were  used  in  B.  J.  until  1875  by  Powalky 
and  Becker.  In  1873  the  correction  to  the  ephemeris  had  increased  to 
—4s  in  a  and  —20"  in  8. 

In  B.  J.  1876  E.  Becker  published  a  preliminary  set  of  elements  which 
finally  was  corrected  to  the  Elements  E,  given  by  Bauschinger12  based 
upon  the  oppositions  1868,  1869,  1871,  1873  and  1874.  Special  per- 
turbations by  Jupiter  and  Saturn  were  included  and  brought  forward 
to  the  osculation  1898,  December  20. 

The  elements  given  in  Kleine  Planeten  for  1920  are  Becker's  brought 
forward  by  Strehlow  with  Jupiter's  perturbations.  The  mean  motion 
is  corrected  empirically  by  +0"-07  (since  1898,  August  22),  represent- 
ing the  fourteen  oppositions  since  1900  within  ztO™^.  Osculation  1920 
January  0.  Elements  F. 

In  the  mean  time  v.  Zeipel13  had  computed  the  tables  for  the  general 
perturbations  by  Jupiter  according  to  the  method  by  Bohlin  in  the  case 


CELESTIAL   MECHANICS:    LEUSCHNER  29 

of  near  commensurability,  Hecuba  group.  As  a  test  he  applied  his 
tables  to  the  orbit  of  (10)  Hygiea  and  started  with  the  Elements  G  (a) 
from  Zech's  manuscript.  These  were  first  transformed  to  mean  ele- 
ments and  then  compared  with  nine  oppositions  from  1849—1884 
with  the  aid  of  his  tables.  A  least  squares  solution  gave  the  corrected 
Elements  H. 

The  work  of  computing  tables  after  Bohlin's  method  for  the  Hecuba 
group  had  also  been  undertaken  by  A.  0.  Leuschner14  and  an  appli- 
cation to  (10)  Hygiea  was  made  by  Miss  E.  Glancy  and  Miss  S.  H. 
Levy14.  In  order  to  compare  the  results  with  those  of  v.  Zeipel,  Miss 
Glancy15  used  the  Berkeley  tables  and  after  a  comparison  with  nine 
oppositions  between  1849-1884  obtained  the  Elements  I,  starting  with 
the  mean  Elements  G(b).  Further  comparisons  were  made  by  repre- 
senting observations  in  1910,  1914,  1917.  The  residuals  before  solution 
from  Zech's  elements  with  the  Berkeley  tables  were  — 11' to  +10'  in 
the  plane;  from  Zech's  elements  and  v.  ZeipePs  tables  — 24'  to  +5'. 
After  Miss  Glancy 's  solution  the  residuals  are  — 8'  to  +7',  and  after 
v.  Zeipel's  solution  —7'  to  +8';  but  in  1917  they  are  +9'  and  +19' 
respectively,  apparently  in  favor  of  the  Elements  I  and  the  Berkeley 
tables. 

The  residuals  from  Zech's  elements  and  the  Berkeley  tables  ( — 11' 
to  +10')  show  a  decided  periodicity  of  30  years,  thus  pointing  to  the 
influence  of  Saturn.  Later  observations  in  1917  and  1921  are  repre- 
sented much  better  (0'  and  +10')  by  the  Berkeley  tables  and  Zech's 
elements  than  by  any  of  the  solutions.16  The  best  future  representa- 
tion may  be  expected  from  Zech's  elements  G(b)  and  the  Berkeley 
tables.  The  residuals,  probably  chiefly  due  to  Saturn,  keep  within 
fixed  limits  ±10'.  The  most  obvious  next  step  would  be  to  correct 
the  residuals  for  some  of  the  earlier  oppositions  by  means  of  the  per- 
turbations of  Saturn  and  Mars,  available  in  manuscript  in  the  Rechen- 
institut.  Until  that  shall  have  been  done,  no  corrections  should  be 
applied  to  Zech's  elements,  which  appear  to  be  the  best  available. 

REFERENCES 

*A.  N.  vol.  28,  p.  391.  "H.  v.  Zeipel,  Angenaherte  Jupiter- 

2  A.  N.  vol.  29,  p.  15.  storungen     f iir     die     Hecuba-Gruppe. 

3  A.  N.  vol.  29,  p.  49.  Memoires  de  TAcademie  des  Sciences 
*A.  N.  vol.  29,  p.  81.  de   St.   Petersburg,   vol.   12,   Nr.   11, 

6  A.  N.    vol.  29,  p.  81.  iqno 

6A.  N.    vol.  29,  p.  81,  p.  126;  vol.  30,         „'    ..      .  ,    a  . 

p   320.  "National     Academy     of     Sciences. 

7  A.  N.    vol.  30,  p.  81,  p.  82.  Memoirs,  vol.  14,  third  memoir. 

8  A.  N.    vol.  30,  p.  87.  15  A.  J.    vol.  32,  p.  27. 

».f"2'    V0}'  81>  P*  27£*  "A>    °'    Leuschner»    Comparison   of 

"  A '  N         1   ^Q  P'  -U7  theory  with  observation  for  the  minor 

"VerSffentUchungen  des  astronomis-  Planets  <10>  Hy«iea  and  <175>  Andr°- 
chen  Recheninstituts  zu  Berlin.  Nr.  16,  mache.  Proc.  N.  A.  S.,  Washington, 
p.  48.  vol.  8,  No.  7,  p.  170. 


30 


CELESTIAL   MECHANICS:   LEUSCHNER 
TABLE  5. — Elements — (10)  Hygiea 


Letter 

Date 

M.T. 

M 

V 

fl 

i 

A.  .      . 

1849  Apr.  15  0 

Berlin.   .  . 

330  52     8.56 

227  49  54.23 

287  37     8.64 

3  47  15  51 

B  

1851  Sept.  28.5 

Berlin  

128  44  20.7 

228     2  32.1 

287  39     8.3 

3  47     8.3 

C  
D  

1851  Sept.  17.0.  . 
1851  Sept.  17.0.. 

Berlin  
Berlin  

126  59  37.2 
126  59  48.76 

227  48     9.2 
227  47  58.77 

287  38  37.6 
287  38  34.21 

3  47    9.22 
3  47     9.29 

E 

1874  Dec    26  0 

Berlin 

174  55  300 

CO 

312  40  30  5 

285  18  57  5 

3  47  43  2 

F      . 

1925  Jan       0  0 

Greenwich 

181  38  49  2 

305  25  22  8 

285  52  55  2 

3  48  46  8 

G(a).. 
G(b).. 
H  

1851  Sept.  17.0.  . 
1851  Sept.  17.0.  . 
1851  Sept.  17.0 

Berlin  
Berlin  
Berlin.... 

126  59  48.6 
121  51  58 
121  35  27.6 

TT 

227  46  36.6 
230  47  49.6 
231     2     9 

287  37  11.4 
287  37  11.28 
287     8  33  6 

3  47     8.4 
3  47    8.5 
3  47  45  6 

I  

1851  Sept.  17.0  . 

Berlin  

121  31  53.8 

231     4  56.6 

287  27  28  1 

3  47  30.1 

Letter 

Date 

M.T. 

V 

M 

Equinox 

Authority 

A  

1849  Apr.    15.0  

Berlin  

Of                •    , 

5  47  55.37 

634.6406 

1849.0 

d'  Arrest  VI 

B  

1851  Sept.  28.5  

Berlin.... 

5  46  34.7 

634.83504 

Chevallier 

C  
D  
E  
F  
G(a).. 
G(b).. 
H  
I  

1851  Sept.  17.0  
1851  Sept.  17.0  
1874  Dec.   26.0..... 
1925  Jan.      0.0  
1851  Sept.  17.0  
1851  Sept.  17.0  
1851  Sept.  17.0  
1851  Sept.  17.0  

Berlin.... 
Berlin.... 
Berlin.... 
Greenwich 
Berlin.... 
Berlin.... 
Berlin  
Berlin  

5  46  16.8 
5  46  16.57 
6  18  23.7 
6  40  48.0 
5  46  16.8 
6  23    8.9 
6  22     1.2 
6  21  31.0 

634.84564 
634.84912 
636.58673 
638.517 
634.850 
636.8566 
636  849 
636  .  86105 

1851  Sept.  17 
1851  Sept.  17 
1870.0 
1925.0 
1850.0 
1850.0 
1850.0 
1850  0 

Zech 
Zech 
E.  Becker 
Strehlow 
Zech 
Zech 
v.  Zeipel 
Glancy 

CELESTIAL   MECHANICS:   LEUSCHNER  31 

(28)  BtiLLONA 

Discovered  by  R.  Luther1  at  Bilk  near  Diisseldorf,  March  1,  1854. 

Preliminary  elements  were  published  by  Bruhns,2  3  4  Chevallier5  and 
Ruemker,5  Oudemans.6 

From  141  observations  formed  into  five  normal  places  Bruhns7  8 
derived,  originally  using  four  longitudes  and  two  latitudes  correspond- 
ing to  the  first,  second,  fourth  and  fifth  normals,  the  first  reliable 
Elements  A.  As  Elements  A  differ  considerably  from  his  previous 
elements  and  those  of  Oudemans.  He  made  a  comparison  of  ephem- 
erides  which  satisfied  him  regarding  the  correctness  of  Elements  A. 
This  case  is  somewhat  indeterminate  and  small  errors  of  observation 
would  produce  considerable  changes  in  resulting  elements.  From 
Elements  A  Bruhns7  has  published  an  ephemeris  for  1855. 

Further  ephemerides  are  published  by  Bruhns,  among  them  for 
1856,9  1866,10  (a  star  correction11  November  29  by  Engelmann),  for 
1867,12  correction  Aa  —50s,  AS  —  3'.4  by  Tietjen,13  and  for  1871-72.14 

Other  elements  by  Bruhns  are  published  in  the  B.  J.  from  1857  to 
1860. 

The  B.  J.  from  1861  to  1891  contains  new  elements  and  ephemerides 
by  Bruhns  originally  based  upon  the  observations  of  the  first  four 
oppositions  with  perturbations  by  Jupiter,  Saturn,  and  Mars.  Ele- 
ments C.19  Whether  these  elements  are  merely  brought  forward  by 
perturbations  or  contain  corrections  is  difficult  to  determine. 

From  1892  the  B.  J.  uses  von  der  Groeben's  elements  instead  of 
those  by  Bruhns. 

By  a  process  of  successive  correction  of  osculating  elements  and 
special  perturbations  by  Jupiter,  Saturn  and  Mars  (perturbations  by 
the  Earth  and  Venus  were  found  negligible) ,  based  on  16  normal  places 
extending  over  a  period  of  thirty-two  years  from  1854  to  1886,  von 
der  Groeben,15  starting  with  a  set  of  elements  osculating  for  1870, 
September  18,  derived  Elements  Ba,  Bb,  Be,  Bd  osculating  for  differ- 
ent epochs  from  1861  to  1882.  These  elements  were  brought  forward 
with  special  perturbations  of  Jupiter,  Saturn  and  Mars  to  the  epoch 
1886,  February  26,  Elements  Be,  and  to  the  epoch  1889,  October  28, 
Elements  Bf.  Elements  Bf  are  adopted  by  the  B.  J.  for  189216  and 
1893.17  Three  observations  by  Ball  at  Luttich,  October  31  to  Novem- 
ber 15,  1889,  are  well  represented  by  the  ephemeris.  Mean  Aa  +s.19, 
mean  AS  —  ".5. 

From  seven  observations  at  Washington  and  Tacubaya  with  star 
places  newly  determined  by  Bruns  and  four  observations  by  himself 
at  Diisseldorf,  Luther18  obtained  a  mean  correction  to  the  ephemeris 


32  CELESTIAL   MECHANICS:   LEUSCHNER 

from  von  der  Groeben's  Elements  Bf  brought  forward  by  special 
perturbations  of  Aa  —  Os.42,  AS  +3".4  in  1894. 

The  Elements  Bg20  to  Bu  given  in  the  B.  J.  and  in  Kleine  Planeten 
from  1894  to  1919  are  von  der  Groeben's  elements  probably  brought 
up  to  date  in  the  same  manner  as  before  with  the  special  perturbations 
by  Jupiter,  Saturn  and  Mars  and  without  any  other  correction. 

The  following  is  a  partial  list  of  published  corrections  to  the  B.  J. 
ephemerides  from  von  der  Groeben's  elements: 

1902,  August  7  +      88  +0'  Luther21 

1903,  October  26  +6ra38  +25' 2  Luther22 

1905,  March  1  +  2.77  -  8'3  Iwanowski28 

1906,  June  20  +  8.99  +11.3  Luther24 

1907,  September  7  +  21.68  +1'  23'l  Luther26 

1908,  November  28  -  17.77  +0'    9'0  Luther26 

1916,  August  8  +0m3  +1'  Luther27 

1917,  December  4  +lm4  +7'  Luther28 
1919,  April  1                     -Im8                   +6'  Luther29 

In  a  dissertation  on  the  Jupiter  perturbations  of  the  group  of  small 
planets  whose  mean  daily  motions  are  in  the  neighborhood  of  750", 
D.  T.  Wilson30  gives  an  application  of  the  Hansen-Bohlin  method  to 
the  Jupiter  perturbations  of  this  group.  "The  integration  divisors  for 
certain  values  of  the  integers  n,  r  and  s  become  as  small  as  0.2.  These 
terms  increase  rapidly  as  the  series  advance.  They  were  computed  to 
the  third  power  of  the  eccentricities  and  to  the  fourth  power  of  w.  It 
was  found  that  all  the  terms  of  the  third  and  fourth  powers  of  w  and 
some  of  those  of  the  third  power  of  the  eccentricities  are  negligible 
when  the  eccentricity  of  the  disturbed  planet  does  not  exceed  0.34  and 
when  the  mean  daily  motion  lies  within  the  limits  720"  to  780".  There- 
fore only  those  terms  of  the  third  power  of  the  eccentricities  which  are 
appreciable  within  the  above  limits  have  been  retained.  All  the  secular 
terms  have  been  computed  to  the  fourth  power  of  eccentricities." 

By  means  of  these  tables  the  Jupiter  perturbations  of  Bellona  were 
computed  and  compared  with  the  results  previously  obtained  by 
Hansen's  method  by  Bohlin.31 

The  mass  of  Jupiter  is  taken  as  1 :  1048  in  the  tables  by  Wilson. 

Of  the  three  applications  of  these  tables  by  D.  T.  Wilson  that  of 
Bellona  is  by  far  the  most  interesting.  This  depends  on  the  greater 
proximity  to  the  commensurability  (748")  and  the  cross  position  of 
the  line  of  apsides  to  that  of  Jupiter.  The  inequality  2g-5g'  in  rj8z 
is  the  largest  of  all  and  amounts  to  40'.  But  the  difference  between 
the  coefficients  computed  by  Hansen's  and  Bohlin's  methods  is 
large  (2')  and  the  comparison  with  the  observations  ought  to  decide 
between  the  application  of  these  methods  to  numerous  planets  in  this 
group. 


CELESTIAL   MECHANICS:    LEUSCHNER  33 

REFERENCES 

SA.N.    vol.  38,  pp.  Ill,  143.  18A.  N.    vol.  139,  p.  302. 

3  A.  N.    vol.  38,  p.  155.  M  B.  J.    1861,  p.  506. 

8  A.  N.    vol.  38,  p.  218.  "  B.  J.    1894,  p.  394. 

4  A.  N.    vol.  38,  p.  351.  31  A.  N.    vol.  159,  p.  295. 
6A.N.    vol.  38,  p.  158.  MA.  N.    vol.  163,  p.  383. 
6A.N.    vol.  38,  p.  158.                                    "A.  N.    vol.  167,  p.  319. 
TA.N.    vol.  40,  p.  203.  *A.  N.    vol.  171,  p.  349. 
8B.  J.    1858,  p.  407.                                       a°  A.  N.    vol.  175,  p.  401. 

9  A.  N.    vol.  44,  p.  235.  M  A.  N.    vol.  179,  p.  243. 

10  A.  N.    vol.  68,  p.  125.  a7  A.  N.    vol.  203,  p.  164. 

11  A.  N.    vol.  69,  p.  104.  M  A.  N.    vol.  206,  p.  16. 
"  B.  J.    1869.  *  A.  N.    vol.  208,  p.  248. 

13  A.  N.    vol.  65,  p.  176.  80  Astrpnomiska  lakttagelser  och  Un- 

14  A.  N.    vol.  79,  p.  5.  dersokningar    a    Stockholms    Observa- 

15  A.  N.    vol.  123,  p.  369.  torium.    vol.  10,  No.  1. 

16 B.  J.    1892,  p.  390.  "Manuscript    in    the    office    of    the 

17  B.  J.    1893,  p.  390.  Recheninstitut. 


34 


CELESTIAL   MECHANICS:   LEUSCHNER 


iij i ijijf jjifji j jjjjj 

3  o  o  o  o  o  a  o  o  o  o  o  a  o  o  o  o  o  o  o  o  o  § 


o   qqqqqqqqqqqqqqqqqqqqq   q 


«D  00 

rH    CO    CO  r-J 

OQ    rH    ,-H  ^ 


t>-          iO 

l2       S 


•*          i-tTjtOSOCOOi^OOcOCOiOOr-lO^iCINt^fNlO  O 

»O  iH          iOiH(NiO<N»O  «OO  i-ii-iCOCO  •* 

t- 


?§        S^^Sc^csS?5^^fe?§2^^S^^M§  ^| 


00         O  O  to  00  O  O5 


2^OOr-(i-lOOOO<»00«OOOOO          •* 
CMCOCOCO^TjtiOlOTjtCOfNr-iO  <N 


(Nb-QCOrH^QOOO 


00   M    •*   <N          r^ 


®       9  d  °.  q  q  o  q'  9  q  q 


Oq'PqqCjq'q'q'o'io       q 
IlllIIlllll      Q 


o!  XI    O  73    9  +6..1IM  -m  *fM  *1    C    a    O    O 

PQ  «  «  W  «  pq  «  pq  M  pq  pq  pq  pq  «  pj  pq 


CELESTIAL   MECHANICS:   LEUSCHNER  35 

(93)  MINERVA 

Discovered  by  J.  Watson  1  1867,  August  24,  at  Ann  Arbor,  and  ob- 
served on  three  successive  days.  Estimated  to  be  of  llth  magnitude. 

P.  Lehmann 2  in  Berlin  computed  his  first  orbit  from  the  observa- 
tions to  October  2,  covering  an  heliocentric  arc  from  60°  to  75°  after 
perihelion.  An  ephemeris  for  1867,  November  and  December, 
deviated  +08.33  in  a,  — 2".0  in  8  from  the  observations.  These  ele- 
ments were  repeated  in  B.  J.  1870.  Elements  A. 

Lehmann  changed  his  Elements  A  to  the  Elements  B  given  in  B. 
J.  1871  without  statement  of  the  observations  used,  the  perturbations 
applied  or  the  representation  of  the  observations.  The  most  prom- 
inent changes  are  the  improvement  of  the  longitude  of  perihelion  by 
1°,  and  also  the  increase  of  the  eccentricity,  besides  diminution  of  the 
mean  motion  by  1"  a  day,  to  a  value  as  much  below  the  mean  mean 
motion  as  the  osculating  value  was  above.  This  error  in  the  mean 
.motion  made  a  new  computation  necessary  which  Lehmann  published 
in  B.  J.  1872  without  explanation.  Elements  C.  A  further  improve- 
ment was  obtained  by  the  next  set  of  Elements  D  by  Lehmann  in 
B.  J.  1873.  These  elements  were  (perhaps)  brought  up  with  per- 
turbations to  the  new  osculation  in  1872  as  published  in  B.  J.  1874 
and  1875.  Elements  E.  To  the. elements  M,  \L  and  <£  corrections  were 
applied.  The  resulting  Elements  F  are  given  in  B.  J.  for  1876-1880. 
Then  Lehmann  undertook  to  make  a  final  determination  of  the  orbit 
from  the  first  7  oppositions  with  special  perturbations  by  Jupiter  and 
Saturn,  (as  stated  in  Bauschinger,  Tabellen,  etc.)  These  Elements  G 
were  probably  published  in  B.  J.  with  change  of  osculation  until  the 
issue  1913. 

Berberich  corrected  the  Elements  G  empirically  by  the  observations 
in  the  oppositions  1899,  1902,  1907,  1908,  1911,  and  derived  the. Ele- 
ments H,  which  were  published  in  B.  J.  1914.  B.  J.  1915  gives  the 
elements  by  Leuschner.6 

The  general  perturbations  by  Jupiter  were  developed  by  W.  S. 
Eichelberger.3  The  basic  elements  were  obtained  "from  a  special 
discussion,  by  the  author,  of  the  observations  from  1867  to  1879,  in- 
clusive." Elements  I.  The  method  is  that  of  Hansen  for  absolute 
perturbations  of  the  first  order  retaining  the  eccentric  anomaly  in 
the  argument.  The  constants  of  integration  were  determined.  With 
these  perturbations  and  the  preliminary  Elements  I  the  observations 
from  1867  to  1884  were  compared.  A  least  squares  solution  was  made 
and  the  final  Elements  J  obtained.  The  comparison  with  the  observa- 
tions is  not  given. 


36  CELESTIAL   MECHANICS:    LEUSCHNER 

Observations  by  T.  J.  J.  See  in  1899  Sept.  compared  with  a  man- 
uscript ephemeris  by  Eichelberger  gave  the  corrections  +0S.8  in  a, 
+50"  in  8.  "The  comparison  would  indicate  that  Eichelberger's 
theory  is  very  good."  Same  year  and  date  Coddington  observed 
Minerva  and  compared  "with  an  ephemeris  furnished  by  Professor 
Newcomb."  Same  corrections  as  See  found  to  Eichelberger's 
ephemeris. 

The  planet  discovery  1902  HQ  (February  25),  by  Wolf4  was  sus- 
pected to  be  (93)  Minerva  and  the  identity  was  confirmed  by  two 
observations  from  Bordeaux.5 

The  work  of  Eichelberger  was  undertaken  originally  under  the 
auspices  of  the  Watson  Trustees,  but  the  Trustees  suspected  that  the 
small  residuals  which  resulted  from  a  comparison  of  theory  and  obser- 
vation were  due  to  some  error.  Investigation  of  the  sources  of  the 
suspected  error  undertaken  by  Leuschner  at  the  request  of  the  Trustees 
confirmed  Eichelberger's  work,  the  representation  of  the  observations 
being  found  entirely  satisfactory,  in  view  of  the  fact  that  only  first 
order  perturbations  of  Jupiter  were  considered. 

Leuschner6  revised  the  elements  by  including  in  the  least  squares 
solution  further  oppositions  to  1902,  so  that  his  elements  are  based 
on  oppositions  extending  from  1867  to  1902.  The  elements  differ  only 
slightly  from  those  by  Eichelberger.  In  the  Perturbations  and  Tables 
of  Twelve  Watson  Asteroids,6  Eichelberger's  perturbations  are  retained 
without  change.  Observations  of  recent  years  are  well  represented  by 
the  ephemerides  published  in  Kleine  Planeten  by  the  Berlin  Rechen- 
institut  on  the  basis  of  these  elements  and  tables,  as  is  indicated  by 
comparison  with  approximate  photographic  positions  at  Konigstuhl7 
in  1918  and  at  Algiers  in  1921.8  Further  corrections  of  the  elements 
should  be  undertaken  only  on  the  basis  of  perturbations  by  Jupiter  of 
the  second  order,  and  of  perturbations  by  other  major  planets. 

For  the  group  of  minor  planets  having  a  mean  motion  of  about  750" 
(the  Minerva  group),  D.  T.  Wilson9  has  computed  tables  after  the 
method  of  Bohlin.  No  application  of  this  theory  seems  to  have  been 
made  to  (93). 

REFERENCES 

1  A.  N.    vol.  70,  p.  45.  *  Memoirs  of  the  National  Academy 

9  A.  N.    vol.  70,  p.  205.  of  Sciences,    vol.  x,  seventh  memoir. 

*  Memoirs  of  the  National  Academy      Washington,  1910. 

of  Sciences,     vol.  viii,  third  memoir.  T  Eph.  Z,  A.  N.    1918/558. 

Washington,  1899.  8C.  O.  M.  1921/171. 

*  A.  N.    vol.  158,  p.  95.  9  Astronomiska  lakttagelser  och  Un- 
'A.  N.    vol.  158,  p.  175;  A.  N.    vol.  dersokningar.     vol.  10,  No.  1,  Stock- 

160,  p.  352.  holm. 


CELESTIAL   MECHANICS:   LEUSCHNER 
TABLE  7.— Elements— (93)  Minerva 


37 


Letter 

Epoch 

M 

7T 

Q 

i 

A  

1867  Oct.    2.0     .  Berlin  

66  47  58  8 

276  39  54.8 

5  2  28  0 

8  35  34  9 

B 

1867  Oct     20        Berlin 

67     1  59  1 

275  38  16  3 

5  4  11  4 

8  36  31  8 

C  

1870  May   1.0...  .Berlin  

270  51  42.1 

275     2  55.0 

5  4  16.2 

8  36  17.6 

D 

1870  May  1  0         Berlin 

270  53  55  4 

275    0  36  8 

5  4  26  8 

8  36  34  6 

E  
F  
G..    .    . 

1872  Nov.  6.  0....  Berlin  
1872  Nov.  6.  0....  Berlin  
1879  Feb     3  0        Berlin. 

109  32  42.8 
109  32  48.4 
241     7  28  6 

274  43  34.4 
274  43  34.4 
274  42  35  8 

5  3  40.3 
5  3  40.3 
5  9  11  9 

8  36  34.3 
8  36  34.3 
8  36     32 

H     . 

1911  Jan.    31  5       Berlin 

236  37  30 

277  36  31  2 

5  4  31  2 

8  35  28  0 

I  

1872  Nov.  2  0        Greenwich 

108  37  48  4 

274  47  41  4 

5  5  25  0 

8  36  21  6 

J  

1872  Nov    2.0        Greenwich 

108  28  35  7 

274  49  19  2 

5  5  17  5 

8  36  23.6 

K  

1875  Jan.    0  0        Greenwich 

278  32     8 

274  51  41 

578 

8  36  20 

Letter 

Epoch 

<P 

M 

Equinox 

Computer 

A 

1867  Oct  2  0  Berlin. 

Off 

7  39  29  5 

» 
776  43667 

1867  0 

Lehmann 

B  
C  
D  
E  
F  

1867  Oct.  2.0.  .  .  .Berlin  
1870  May  1.0  Berlin  
1870  May  1.0  Berlin  
1872  Nov.  6.0  Berlin  
1872  Nov.  6  0  Berlin. 

8    3  47.9 
8    3  55.5 
8    4  38.9 
8    4  43.5 
8    4  45.1 

775.5500 
776.41806 
776.51030 
776.47953 
776.49465 

1870.0 
1870.0 
1870.0 
1870.0 
1870.0 

Lehmann 
Lehmann 
Lehmann 
Lehmann 
Lehmann 

G  
H  
I  
j 

1879  Feb.  3.0.  .  .  .Berlin  
1911  Jan.  31.  5...  Berlin  
1872  Nov.  2.0  Greenwich  
1872  Nov  2  0  Greenwich 

7  59     4.8 
8  15  30 
8    5     0.5 
8    4  52  4 

775.63887 
775.6316 
776.51130 
775  920408 

1880.0 
1910.0 
1872.0 
1872  0 

Lehmann 
Berberich 
Eichelberger 
Eichelberger 

K  

1875  Jan.  0.0.  ..  .Greenwich  

8    4  54 

775.9214 

1875.0 

Leuschner 

38  CELESTIAL   MECHANICS:    LEVSCHNER 

(94)   AURORA 

Discovered  by  Watson1,  1867,  September  6,  at  Ann  Arbor. 

Preliminary  elements  were  computed  by  Tietjen23,  the  second  set 
given  below  as  Elements  A. 

With  Elements  A,  H.  Leppig4  formed  eight  normal  places  over 
162  days,  and  determined  Elements  B.  An  accurate  ephemeris  includ- 
ing special  perturbations  by  Encke's  method  was  computed  for  1868, 
December  15  to  1869,  January  31.  Observations  by  Vogel,  January 
15,  17,  18,  1869,  gave  residuals,  Aa  — 21S.6,  AS  +53".5. 

Thereafter,  the  B.  J.  gives  elements  by  Leppig  from  1872  to  1915, 
(but  with  a  misprint  of  4°  in  o>  from  1887  to  18975),  Bauschinger 
gives  Leppig's  Elements  C  in  the  "Tabellen"6  for  the  epoch  1883,  July 
12.0.  The  various  elements  in  the  B.  J.  to  1915  are  probably  brought 
up  from  Leppig's  Elements  B,  with  special  perturbations.  The  char- 
acter of  the  perturbations  is  not  given.  Nor  is  any  reference  made 
to  arbitrary  corrections. 

In  Kleine  Planeten,  1916,  the  elements  are  changed  by  estimating 
the  perturbations  1883-1910  and  roughly  determining,  M  and  /*,,  Ele- 
ments D. 

In  Kleine  Planeten,  1921,  the  last  elements  are  again  corrected  by 
the  computation  of  Jupiter's  perturbations  and  a  representation  of 
observed  positions,  1884-1918,  within  ±lm.5.  Osculating  elements, 
1921,  April  24,  not  available. 

From  1884  to  1899  the  planet  was  practically  lost,  mainly  on  ac- 
count of  the  misprint  in  to.  Coddington7  computed  a  place  with  the 
elements  of  B.  J.  1901,  and  found  the  planet  Aa  +5m.2,  AS  —20', 
1899.  For  the  computation  of  the  perturbations  and  tables  of  the 
Watson  asteroids,  Leuschner  made  a  collection  of  Leppig's  elements  in 
the  B.  J.  including  Elements  C  and  derived  average  Elements  E  from 
those  in  B.  J.  1871,  1873,  1874,  1875,  1877,  1878,  1881,  1898.  From 
these,  approximate  mean  elements  were  derived.  The  perturbations 
were  developed  and  a  preliminary  correction  of  the  mean  motion 
and  mean  anomaly  from  observations  in  1867  and  1899  was  at- 
tempted. A  mistake  in  one  of  the  main  terms  of  the  perturbations 
was  discovered  and  corrected.  Nevertheless,  large  discrepancies  be- 
tween observation  and  computation  remained  ( — 5°  in  a,  1867).  These 
differences  were  used  for  a  preliminary  correction  of  the  mean  motion 
and  of  the  mean  anomaly.  With  the  new  value  of  the  mean  motion 
the  perturbations  were  corrected,  the  residuals  re-determined,  and 
further  corrections  made  to  the  mean  motion  and  to  the  mean 
anomaly.  A  least  square  solution  of  12  places  from  1867  to  1899 
was  then  made  including  the  corrected  perturbations.  The  residuals 


CELESTIAL   MECHANICS:   LEUSCHNER 


39 


were  thereby  reduced  from  ±2°.2  to  a  maximum  of  dbO°.14.  Ele- 
ments F. 

Further  correction  of  the  perturbations  by  means  of  the  new 
mean  motion  produced  larger  residuals.  The  best  representation  as 
above  is  obtained  by  the  use  of  the  adopted  Elements  F  without 
further  correcting  the  perturbations. 

From  the  experience  of  the  Recheninstitut  and  of  Leuschner 
with  Leppig's  elements,  it  is  evident  that  before  a  satisfactory  repre- 
sentation for  all  oppositions,  without  making  arbitrary  corrections  to 
the  elements,  can  be  obtained,  it  will  be  necessary  to  derive  an  ac- 
curate set  of  osculating  elements  by  connecting  a  limited  number  of 
oppositions  with  accurate  determination  of  the  perturbations.  With 
an  accurate  set  of  osculating  elements  the  perturbations  may  then 
be  corrected,  but  further  correction  of  the  elements  should  be  made 
only  after  higher  order  perturbations  and  perturbations  by  planets 
other  than  Jupiter  shall  have  been  considered. 


REFERENCES 


'A.  N. 
2  A.  N. 
8  A.  N. 
4B.  J. 


vol.  70,  p.  79. 

vol.  70,  p.  219. 

vol.  71,  p.  47. 

1871.    A.  N.  vol.  72,  p.  331. 


8  A.  N.    vol.  139,  p.  63. 
*Tabellen  zur  Geschichte  und  Stat- 
istik  der  Kleinen  Planeten. 
7  A.  N.    vol.  153,  p.  225. 


TABLE  8.— Elements— (94)  Aurora 


Letter 

Epoch 

M.T. 

M 

0> 

D 

i 

A.  .  .    . 

1867  Nov.  28  0 

Berlin    .  . 

340  30  39  5 

40  50  23  2 

4  32    9  3 

8  5  27  0 

B...    . 

1870  Jan.  O.O.... 

Berlin  

115    9  43.74 

40     2  43.08 

4  34  36.38 

8  5  18.49 

C...    . 

1883  July    12.0.. 

Berlin  

256     3     4.3 

45  22  31.8 

4  25     0.9 

8  4  14.0 

D...    . 

1925  Jan.  0.5  

Greenwich  . 

20    40     1.2 

57  20     9.6 

4  22  26.4 

8  4     8.4 

E...    . 

1875  Jan.    0.0... 

Greenwich  . 

73    37  45 

41  51  21 

4  28  46 

8  4  54 

F...    . 

1875  Jan.    0.0... 

Greenwich. 

65     20  12 

48  13  54 

4  24  21 

8  3  51 

Letter 

Epoch 

M.T. 

9 

M 

Equinox 

Author 

A 

1867  Nov    28  0 

Berlin    .  . 

Of                W 

5  10  18  6 

630  5129 

1867  0 

Tietjen 

B 

1870  Jan       0  0 

Berlin  ..      . 

5     6     8  13 

631  5264 

1870  0 

Leppig 

c 

1883  July    12  0 

Berlin  

4  44  18  3 

630.6584 

1900.0 

Leppig 

D 

1925  Jan.      05. 

Greenwich.  . 

5    4  58.8 

631.800 

1925.0 

Berberich 

E  

1875  Jan.      00     

Greenwich.  . 

4  56  22 

631.2196 

1900.0 

Leuschner 

F  

1875  Jan.      0.0  

Greenwich.  . 

5  17  16 

631.9473 

Leuschner 

40  CELESTIAL  MECHANICS:   LEUSCHNER 

(127)  JOHANNA 

Discovered  by  Prosper  Henry1  at  Paris,  1872,  November  5. 

In  B.  J.  1875  a  set  of  Elements  A  by  Baillaud  is  published,  based  on 
observations  Nov.  9,  22,  28. 

Preliminary  Elements  B  by  Renan2  are  based  on  seven  observations 
for  the  opposition  1872-73.  An  ephemeris  for  1874  April  and  May 
is  also  published.  An  observation  on  April  17,  1874,  shows  corrections 
to  the  ephemeris  Aa  +2m43s,  AS  —16'. 

An  improvement  on  Elements  B  is  made  by  Renan3  on  the  basis  of 
six  normal  places  (1872-1874).  The  resulting  Elements  C  represent 
the  normal  places  within  +7".8  and  — 9".l.  Renan  states  these  differ- 
ences are  within  the  limits  of  error  and  Elements  C  may  be  considered 
definitive.  An  ephemeris  is  then  computed  for  1876  September. 

Elements  D  are  published  by  Bauschinger.4  They  are  by  Maywald 
and  are  based  on  oppositions  1872,  1874,  1876,  1879.  The  special 
perturbations  of  Jupiter  and  Saturn  to  1890  are  included. 

Renan's  elements  are  published  and  used  by  B.  J.  1882  to  1887. 
Beginning  with  B.  J.  1888,  Maywald's  elements  are  used.  An  empirical 
correction  was  applied  to  the  mean  anomalie  in  1913  by  Berberich.5 

A  correction  to  the  R.  I.  ephemeris6  for  1918,  November  23,  is  Aa 
— 5m.2,  AS  — 20'.  An  improved  ephemeris  is  published7  for  the  period 
1920,  March  21  to  April  21,  from  elements  based  on  6  oppositions, 
1897-1908.  Jupiter's  perturbations  are  included.  Representation 
±0m.3.  Osculation  1921,  July  13.  The  correction  to  this  ephemeris  8 
from  an  observation 8  February  28,  1920,  is  Aa  ^-O'M,  AS  — 6'. 

Another  ephemeris  9  for  1920  by  P.  Maitre  is  based  on  elements 
published  in  Connaissance  des  Temps  for  1915  with  the  mean  anomaly 
corrected,  on  the  basis  of  observations  in  1917  and  1918,  AM  — 1°.305. 

General  perturbations  applying  Bohlin's  method  for  this  planet 
have  been  published  by  D.  T.  Wilson10  and  similar  perturbations  with 
Hansen's  method,  by  M.  Viljev.11 

Olson12  has  published  general  perturbations  of  the  first  order  by 
Jupiter.  The  basic  elements  are  those  of  Maywald.  Jupiter  mass 
1:1047.568  (Bessel-Schur) .  Method  that  of  Hansen.  Terms  of  6th 
to  8th  order  are  below  1".  No  comparison  with  observations. 

REFERENCES 

'A.  N.    vol.  80,  p.  239.  7B— Z  der  A.  N.    No.  11,  1920. 

aA.  N.    vol.  83,  p.  349:  C.  R.    vol.  8B— Z  der  A.  N.    No.  14,  1920. 

78,  1874,  p.  1219.    '  8  C.  O.  M.    No.  297. 

8C.  R.    vol.  83,  1876,  p.  567.  10Ast.  lakttagelser  och  Undersoknin- 

4  Veroffentlichungen  R.  I.  No.  16.  gar,  Stockholm,    vol.  10,  No.  1. 

6B.  J.    1915,  p.  27.  "Bull.  Soc.  Astr.  Russio.    No.  22. 

*A.  N.    vol.  208,  p.  14.  "Swed.    Akad.    Handl.,    Stockholm, 

1895. 


CELESTIAL  MECHANICS:   LEUSCHNER 
TABLE  9.— Elements— (127)  Johanna 


41 


Letter 

Epoch 

M.T. 

M 

<•> 

fl 

i 

A  

1872  Dec.    18  0 

Berlin 

293    6  46 

90  25  17 

31  40  11 

8  19  42 

B  

1874  Apr.    17  0     . 

Berlin 

35     3  42 

91  13  48 

31  41  41 

8  17  28 

C  

1876  Sept     5  5 

Berlin 

223  47  46 

90  50  37 

31  46  38 

8  16  40 

D  

1879  Apr.     4.0 

Berlin 

67  49  52 

89  18  47 

31  45     2 

8  16  48 

Letter 

Epoch 

M.T. 

<P 

M 

Equinox 

Authority 

A.  .    . 

1872  Dec     18  0 

Berlin 

or* 

4  36  31 

» 

766  23 

1872  0 

Baillaud 

B  

1874  Apr     17  0 

Berlin 

3  35  48 

776  368 

1874  0 

C  

1876  Sept     5  5 

Berlin 

3  46  51 

775  9173 

1880  0 

D  

1879  Apr      4  0 

Berlin 

3  51  17 

775  7686 

1880  0 

Maywald 

42  CELESTIAL   MECHANICS:   LEUSCHNER 

(128)  NEMESIS 

Discovered  by  Watson1  at  Ann  Arbor  on  November  25,  1872,  and 
also  by  Borrelly2  at  Marseilles  on  December  4,  1872. 

Preliminary  Elements  A  were  computed  by  Bossert,3  based  on  obser- 
vations 1872,  November  25,  December  7,  and  December  22.  He  also 
published  a  short  search  ephemeris. 

Preliminary  Elements  B  based  on  observations  covering  the  first 
five  months  were  published4  by  Leo  de  Ball.  He  then  forms  six  normal 
places  from  the  same  series  of  observations  and  from  these  determines 
Elements  C.  These  elements  represent  the  normal  places  within 
— 2".8  and  +3".2.  An  ephemeris  for  1874  is  given  in  the  same 
reference,  page  374. 

Preliminary  Elements  D,  based  on  observations  1872,  November  25, 
December  4,  and  December  12,  are  published  by  H.  Richter.6 

Elements  E,  based  on  eight  normal  places  from  the  first  two  oppo- 
sitions (1872-1874)  were  computed  by  Ball.6  The  normal  places  are 
represented  in  the  plane  between  +2".  13  and  — 2".54  and  perpen- 
dicular to  the  plane  between  +2".7  and  — 2".3.  The  special  per- 
turbations of  Jupiter  and  Saturn  were  taken  into  consideration. 

Further  Elements  F  for  a  new  epoch  and  mean  equinox,  in  which 
the  special  perturbations  due  to  Jupiter  and  Saturn  have  been  taken 
into  account,  have  been  published  by  Ball8  and  also  an  ephemeris 
for  1875. 

The  maximum  correction  to  the  ephemeris7  computed  from  Ball's 
elements,  for  July  1880,  is  Aa  +23.14  and  AS  +12".0. 

Elements  G  by  Palisa  are  published  and  used  in  B.  J.  1883,  to  B.  J. 
1893.  They  are  based  on  5  oppositions  1872-79  and  include  the 
perturbations  by  Jupiter  and  Saturn. 

Ball's  elements  are  again  published  and  used  in  B.  J.  1894  (see 
Elements  H)  to  B.  J.  1913. 

Empirical  corrections 8  were  applied  to  Ball's  Elements  H,9  by 
Berberich. 

The  most  extensive  investigation  on  this  planet  is  by  Leuschner.10 
The  Elements  I  are  based  on  observations  extending  from  1872  to  1899 
and  include  the  general  perturbations  due  to  Jupiter  of  the  first  order. 
These  elements  and  perturbations  are  used  in  the  B.  J.  1915  and  to 
date. 

The  correction  to  the  ephemeris11  based  on  Elements  I  on  June  16, 
1921,  was  Aa  +0m.6  and  AS  — 1'.  Corrections  to  these  elements 
should  be  applied  only  on  the  basis  of  higher  order  perturbations  by 
Jupiter  and  of  perturbations  by  other  planets. 


CELESTIAL  MECHANICS:   LEUSCHNER 


43 


REFERENCES 


"A.  N.    vol.  80,  p.  319. 
aA.  N.    vol.  80,  p.  297. 
'A.  N.    vol.  80,  p.  351. 
4  No.  5,  Circular  der  Berliner  Stern- 
warte.    A.  N.    vol.  82,  p.  281. 

8  A.  N.    vol.  82,  p.  63. 

9  A.  N.    vol.  85,  p.  331;  vol.  86,  pp. 
29-31. 


TA.  N.  vol.  98,  p.  55,  and  vol.  99, 
p.  251. 

8  A.  N.  vol.  189,  p.  173. 

»B.  J.  1914. 

10  Mem.  of  the  National  Academy  of 

Sciences,  vol.  x,  seventh  mem.,  p.  255. 

11 C.  0.  M.    No.  181. 


TABLE  10.— -Elements— (128)  Nemesis 


Letter 

Epoch 

M.  T. 

M 

CO 

n 

i 

A..... 
B  

1873  Jan.    1.0 
1873  Feb.  25.5 

Greenwich 
Berlin  

49  56  20 
59  55  44 

296     1  28 
298  35  26 

76  35  50 
76  39  55 

6  18  27 
6  15    9 

C  

1873  Feb.  25  .  5 

Berlin  

59  57  21 

298  34  00 

76  40  02 

6  15  14 

D  

1872  Nov.  25.0 

Berlin  

62  39  10 

274    9  56 

75  12  46 

7  30  40 

E  

1873  Feb.  25.5 

Berlin  

59  52  17 

298  41     4 

76  37  42 

6  15  14 

F 

1875  Apr.  25  0 

Berlin  

228  43  54 

300    3  32 

76  30  40 

6  15  31 

G  

1880  July    7.0 

Berlin  

279  16  11 

300  16    4 

76  32    4 

6  15  43 

H  
I  

1892  Feb.  15.0 
1896  July   3.0 

Berlin  
Berlin  

116  23  35 
101  41     9* 

299  30  58 
299  56  32 

76  36  54 
76  39  30 

6  15  24 
6  15  18 

M 
Letter 

Epoch 

M.  T. 

<f> 

M 

Equinox 

Authority 

A  

1873  Jan.    1.0... 

Greenwich 

O          I            It 

7  21  59 

// 

776.86 

1873.0 

Bossert 

B  
C  

1873  Feb.  25.5... 
1873  Feb.  25.5... 

Berlin.... 
Berlin  

7  13    5 
7  12  51 

778.030 
778.1516 

1873.0 
1873.0 

De  Ball 
De  Ball 

D 

1872  Nov.  25  0.  . 

Berlin  

7  13  17 

764.990 

1872.0 

Richter 

E  
F  

1873  Feb.  25.5... 
1875  Apr.  25  0... 

Berlin.... 
Berlin  .... 

7  11  59 
7  13  20 

778.0333 
777.4729 

1870.0 
1875.0 

DeBall 
DeBall 

G 

1880  July    70... 

Berlin  

7  21  52 

777.4964 

1880.0 

Palisa 

H  
I  

1892  Feb.  15.0... 
1896  July    3.0... 

Berlin.... 
Berlin  

7  10  53 
7  16  50 

777.6921 
777.8761* 

1890.0 
1900.0 

De  Ball 
Leuschner 

*Mean  Elements. 


44  CELESTIAL   MECHANICS:    LEUSCHNER 

(175)  ANDROMACHE. 

Discovered  by  J.  Watson1  1877  October  1,  but  observed  only  Oc- 
tober 5,  6,  16,  29.  These  were  all  the  observations  available  until 
rediscovery  May  19,  1893. 

By  an  unfortunate  delay  the  news  did  not  reach  other  interested 
observatories  until  two  months  later.  The  preliminary  orbit,  Ele- 
ments A,  by  Tietjen,2  was  erroneous,  partly  on  account  of  errors  in 
Watson's  ringmicrometer  observations. 

Watson3  computed  Elements  B,  and  with  them  perturbations  for 
three  years  until  his  death  in  1880. 

Bidschof 4  made  an  attempt  to  improve  the  orbit,  Elements  C,  but 
perceived  the  impossibility  of  this  undertaking. 

1893,  May  19,  Charlios5  at  Nice  discovered  by  photography  a 
planet,  1893  Z,  and  noticed  the  close  similarity  between  its  orbit  and 
that  of  (175)  Andromache.  He  referred  the  case  to  Berberich6  who 
followed  it  out  from  a  preliminary  orbit  to  the  best  to  be  obtained 
from  the  data  in  1893  May  19- August  1,  Elements  D,  and  compared 
his  results  with  the  observations  in  1877  and  with  one  in  1892  from  a 
photographic  plate  taken  at  Heidelberg. 

Berberich7  next  computed  the  extremely  large  perturbations  by 
Jupiter  (in  1887)  and  Saturn,  and  improved  the  orbit  by  a  solution 
from  four  normal  places  in  1877,  1892,  1893,  with  an  ephemeris  for 
the  coming  opposition,  Elements  E — "this  planet  therefore  deserves 
peculiar  attention  for  it  will  furnish  an  excellent  means  for  determin- 
ing an  accurate  value  of  the  mass  of  the  planet  Jupiter."  Cf.  (3) 
Juno,  (4)  Vesta,  (13)  Egeria,  (24)  Themis,  (33)  Polyhymnia,  (447) 
Valentine. 

Berberich8  improved  his  elements  by  the  following  oppositions  and 
brought  them  forward  with  the  special  perturbations  of  Jupiter  and 
Saturn.  The  Elements  F,  G,  H,  illustrate  the  large  changes  in  this 
case  of  near  commensurability  (Hecuba  group). 

This  case  among  others  impelled  A.  0.  Leuschner  to  undertake  the 
computation  of  the  tables9  for  the  Hecuba  group  after  Bohlin's 
method.  The  application  of  these  tables  to  the  orbit  of  (175) 
Andromache  was  carried  out  by  Miss  S.  H.  Levy.  Her  unpublished 
computation  contains  the  transformation  of  Berberich's  elements  to 
Mean  Elements  I,  tables  of  the  perturbations,  determination  of  con- 
stants, the  comparison  with  ten  oppositions  between  1893  and  1907, 
and  at  least  squares  solution  which  led  to  the  Mean  Elements  J.  The 
representation  of  observations  in  1914  was  — Om.4  in  a  and  +  1'  in  8. 

The  comparison  between  theory  and  observation  has  been  discussed 
by  A.  0.  Leuschner.10 


CELESTIAL   MECHANICS:    LEUSCHNER 


45 


REFERENCES 


1C.  R.  vol.85,  p.  1006. 
91,  p.  127. 


A.  N.  vol. 


Circular  zum  B.  J.  Nr.  81. 
8B.  J.  1881,  1882,  1886. 

4  Sitz-Ber.  Akad.  Wien.  Bd.  I 

5  A.  N.    vol.  132,  p.  367. 
e  A.  N.    vol.  134,  p.  143. 

7  A.  J.     vol.  14,  p.  36. 

8  A.  N.    vol.  153,  p.  59. 


9  National  Academy  of  Sciences,  Me- 
moirs, vol.  14,  third  memoir. 

10  A.  O.  LEUSCHNER,  Comparison  of 
theory  with  observation  for  the  minor 
planets  (10)  Hygiea  and  (175)  Andro- 
mache  with  respect  to  the  perturba- 
tions by  Jupiter.    Proc.  N.  A.  S.  Wash- 
ington,   vol.  8,  No.  7,  p.  170. 


TABLE  11. — Elements — (175)  Andromache 


Letter 

Date 

M.  T. 

M 

CO 

Q 

.     i 

A  .  . 

1877  Oct.  29.5  
1878  Dec.    5.O.... 
1889  Apr.    7.5  
1893  Aug.    1.5.... 
1894  Aug.  23.0.... 
1877  Oct.  11.0.... 
1900  Sept.    1.0... 
1920  Jan.    20  
1877  Oct.  11.0.... 
1877  Oct.  11.0  

Berlin  

45     6  25.6 
105  28  46.7 
317     3  18.5 
297  57  33.9 
3  52  54.5 
32     5    0.1 
16  10  41.5 
76  24  14 
3  49  12 
3  59  32 

269  26  21.1 
269  31  45.6 
269  42     7.7 
299  49     1.8 
299  46     4.9 
298  31  16.6 
301  33     8.5 
306  45  58 
304     6  54  * 
304  13     9 

23  32  56.0 
23  34  50.2 
23  43  24.9 
25  27  14.8 
25  27  32.5 
25  36     3.8 
25  23  37.7 
25     7  12 
25  36    4 
25  43  27 

3  46  38.8 
3  46  36.7 
3  46  45.8 
3  10  51.4 
3  10  59.4 
3  11  42.8 
3  10  38.9 
3  10  37 
3  11  43 
3  11  29 

B  .  . 

Berlin  

C  
D  
E  
F  
G  
H  

Berlin  

Berlin  

Berlin  

Berlin  

Berlin  

1925.0  Greenwich.  . 
Berlin  

I  

J  

Berlin  

Letter 

Date 

M.  T. 

<f> 

M 

Equinox 

Authority 

A 

1877  Oct  29  5 

Berlin  ... 

o      /        » 
20  26  45  7 

* 
542  173 

1877  0 

Tie  t  jen 

B    

1878  Dec     5.O.. 

Berlin     . 

20  26  12  6 

541  779908 

1880  0 

Watson 

c  

1889  Apr.    7  5 

Berlin.   . 

20  15  17  8 

544  411 

1890  0 

Bidschof 

D   

1893  Aug.    1  5  . 

Berlin... 

11  39  45  8 

614  943 

1890  0 

Berberich 

E  

1894  Aug.  23.0  . 

Berlin  ..    . 

11  36  51  4 

614  63354 

1890  0 

Berberich 

F  

1877  Oct   11.0 

Berlin.   . 

12     8  54  1 

617  7375 

1890  0 

Berberich 

*  J^  G  

1900  Sept.    1.0 

Berlin 

11     7  42  9 

612  2868 

1900  0 

Berberich 

H  

1920  Jan.    20  
1877  Oct.  11.0 

1925.0  Greenwich.. 
Berlin  .. 

10  42  11 
12  21  26 

607.899 
619  5629 

1925.0 
1890  0 

Berberich 
Berberich 

J  

1877  Oct.  11.0  .. 

Berlin  

12  19  34 

619  025 

1890  0 

Miss  Levy 

C7 

46  CELESTIAL   MECHANICS:   LEUSCHNER 

(433)   EROS,  1898  DQ. 

Discovered  1898,  August  13,  by  Witt  at  Berlin  and  by  Charlois  at 
Nice. 

Fayet1  computed  Elements  A  from  three  observations  on  August 
15,  26,  and  September  7.  Later  he2  computed  Elements  B  from 
normal  places  1898  August  16.5(7  observations),  September  17.5(6 
observations),  and  October  22.4(2  observations).  The  residuals  vary 
from  —0s.  11  to  — 08.36  in  a,  and  +5".l  to  +8".2  in  8. 

Hussey3  gives  Elements  C,  computed  from  the  mean  of  two  Kiel 
observations  made  on  the  15th  of  August,  and  observations  made  at 
Mt.  Hamilton  on  September  6  and  27.  Hussey4  later  computed  Ele- 
ments D  from  observations  at  Mt.  Hamilton  on  August  15,  September 
27,  and  November  11,  1898.  The  residuals  for  the  observations  in 
1898  vary  from  +0S.04  to  — 08.18  in  a,  and  +2".2  to  +4".4  in  8 
and  observations  to  May  4,  1899,  are  closely  represented. 

Chandler5  computed  Elements  E  from  observations  August  14  to 
November  16,  1898.  Elements  F6  and  G7  are  also  due  to  Chandler. 
Elements  F  are  from  eight  normal  places  from  August  17.5  to  No- 
vember 26.5.  The  representation  is  within  the  errors  of  observation. 

In  a  later  article  he  gives  the  following  residuals  for  earlier  observa- 
tions found  on  plates  taken  in  1896  at  Arequipa: 

1896  Aa  AS 

April  6  -Om  36?0  -4'9 

June  5  -1    16  7  -5'8 

Elements  G7  are  the  preceding  ones  with  corrections  applied  so 
as  to  fit  the  observations  made  at  Arequipa  1896.  Representation8 
for  1893,  1894,  1896,  gives  maximum  residual  of  +78.6  in  a  and 
— I'.O  in  8. 

The  observations  from  1893  to  February  16,  1894,  were  found  by 
Pickering  and  Mrs.  Fleming  on  plates  taken  at  Cambridge.  Per- 
turbations were  not  considered  in  applying  the  corrections  to  Ele- 
ments F  to  obtain  Elements  G. 

Elements  H  are  due  to  Berberich9  and  are  given  under  the  title  of 
"First  elements,"  and  are  based  upon  observations  made  at  Urania 
(Berlin)  on  the  14,  23,  and  31  of  August.  Berberich  gives  forty-three 
sets  of  residuals  covering  the  period  August  13  to  August  31,  Aa  be- 
ing greater  than  Os.40  but  three  times,  and  AS  being  larger  than 
10".0  but  twice. 

Elements  I10  J11  K12  and  L13  are  due  to  Russell.  Elements  I  are 
computed  from  three  normal  places  obtained  by  comparison  of  ob- 
servations in  the  A.  N.  and  A.  J.  with  places  computed  from  Ber- 


CELESTIAL   MECHANICS:    LEUSCHNER  47 

bench's  Elements  H.9  (1898,  August  18.5,  34  observations,  August 
26.5,  16  observations,  and  September  9.5,  26  observations,  heliocentric 
arc  about  8°.)  The  set  J11  is  based  upon  nine  normal  places,  although 
Dr.  Chandler's  value  of  the  mean  motion  was  taken  (Elements  G) 
and  the  other  elements  determined  by  varying  the  ratio  of  the  ex- 
treme geocentric  distances.  The  normal  places  are  well  represented. 

Elements  K,  Russell,12  are  the  same  as  J11  except  for  changes  in 
M,  (o,  and  </>  based  upon  observations  in  1899.0  at  the  Chamberlin 
and  Lick  Observatories.  The  representation  August  17,  1898,  to  May 
20,  1899,  is  satisfactory  except  for  the  normal  place  of  November  11.5 
for  which  Aa  — s.28,  AS  — 3". 

Elements  L  are  merely  the  preceding  ones  brought  up  to  the  epoch 
1900.0  and  mean  equinox  of  1900.0.  In  his  article  13  Russell  develops 
the  general  perturbations  of  the  major  axis  of  Eros  by  the  action  of 
Mars.  He  does  this  by  Le  Verrier's  method  of  interpolation.  Rus- 
sell finds  eight  terms  of  the  general  perturbations  of  the  mean  longi- 
tude larger  than  1".50.  The  largest  is  35"  with  a  period  of  about 
1000  years.  The  greatest  displacement  due  to  the  first  7  terms  will 
be  +38"  in  1927  and  —53"  in  1959  in  mean  longitude,  and  "will 
eventually  lead  to  a  valuable  determination  of  the  mass  of  Mars." 
Russell  then  gives  tables  of  the  perturbative  function,  perturbations 
of  log  a  and  perturbations  of  the  mean  longitude. 

Elements  M  14  were  developed  by  Robbins  from  Elements  G  by 
applying  special  perturbations  of  Venus,  Earth,  Mars,  Jupiter,  and 
Saturn  by  the  method  of  the  variation  of  constants.  (Nautical 
Almanac  1837,  appendix.) 

Elements  N15  are  due  to  H.  Osten,  who  has  computed  eight  normal 
places  based  upon  Elements  P,  with  special  perturbations  of  Venus, 
Earth,  Mars,  Jupiter,  and  Saturn  according  to  Encke's  method. 

Millosevich  has  produced  numerous  sets  of  elements.  Elements  O16 
were  computed  from  observations  made  during  the  interval  August  14 
to  September  21,  1898.  An  observation  by  Millosevich  October  8. 
gives  Aa  — ls.93,  AS  +7".5. 

Elements  P17  were  computed  from  a  normal  place  of  date  August 
14.5  and  Millosevich's  observations  on  September  21  and  October  24, 
1898.  Millosevich  states  that  Berberich's  ephemeris  requires  a  correc- 
tion of  +1318  in  a  and  +5'.5  in  8  on  December  23,  1898. 

Elements  Q18  are  from  photographic  observations  at  Greenwich  by 
the  variation  of  the  distances.  Later19  he  stated  that  there  is  an 
error  of  about  2s  in  his  ephemeris  after  five  months. 

Elements  R19  are  based  upon  four  normal  places  and  are  Elements 
P  improved  by  the  method  of  variation  of  the  distances.  They  rep- 
resent 17  normal  places  from  1000  observations  in  1898-1899  perfectly. 


48  CELESTIAL   MECHANICS:    LEUSCHNER 

Elements  S20  are  based  upon  17  normal  places  made  from  999  ob- 
servations in  a  and  992  observations  in  8  in  the  years  1898-1899,  and 
are  derived  from  Elements  R  brought  forward  with  the  perturbations 
of  Venus,  Earth,  Mars,  Saturn  and  Jupiter  for  20-day  intervals. 

Elements  T21  U22  V23  W24  X25  Y26  are  improvements  of  preceding 
elements,  as  are  Elements  Z27  AA27  AB27  AC28  AD29  and  AE30. 
Elements  AE  are  Elements  AC  with  special  perturbations  for  20-day 
intervals  for  the  period  1901,  March  20,  to  1903,  June  8.0  applied. 
These  perturbations  were  computed  by  Wedemeyer,  those  of  Venus, 
Earth,  Mars,  Jupiter  and  Saturn  being  considered. 

Elements  AF  were  found  only  in  the  B.  J.  for  1907.  They  are 
probably  Elements  AE  with  the  epoch  changed  and  perturbations  ap- 
plied. Dziewulski  has  published  "Sekulare  Marsstorungen  in  der 
Bewegung  des  Eros.,"  Bull,  de  PAcad.  des  Science  de  Cracovie  1905. 
Not  available  here.  Some  mistakes  are  corrected  in  A.  N.  vol.  175, 
p.  171.  In  A.  N.  vol.  175,  p.  17,  Merfield  gives  the  secular  perturba- 
tion of  Eros  due  to  all  major  planets.  Gauss'  method  by  Hill. 
Elements  from  Hill's  memoir,  Ast.  Papers  of  the  Amer.  Ephm.  Vol.  IV 
and  Elements  W.  The  secular  perturbations  by  Jupiter,  the  Earth  and 
Venus  are  the  largest. 

Elements  AG  are  by  G.  Witt.31  They  are  based  upon  observations 
from  1893  to  1903,  the  perturbations  of  Venus,  Earth,  Mars,  Jupiter 
and  Saturn  being  included.  The  perturbations  were  calculated  by  the 
method  of  variation  of  constants.  For  Mars,  Jupiter  and  Saturn  the 
perturbations  were  calculated  for  20-day  intervals  and  for  Venus  and 
the  Earth  at  10-day  intervals.  With  these  elements  he  calculates  the 
perturbations  from  1903  to  the  beginning  of  1908,  and  includes  these 
in  an  ephemeris  for  1905  for  dates  from  July  17  to  August  22 
(B.  J.  1907,  p.  476.). 

Elements  AH32  AI33  AJ34  are  preceding  elements  with  change  of 
osculation.  Elements  AI  are  also  found  in  the  Connaissance  des 
Temps  for  1915,  but  with  the  equinox  changed  in  1920.0,  Greenwich 
M.  T. 

In  an  article  "Beitrage  zur  Theorie  der  Bewegung  des  Planeten  433 
Eros,"  E.  Noteboom35  uses  Elements  AL  to  compute  the  general 
perturbations  of  Mercury,  Uranus,  and  Nepture.  A  recurring  run 
in  the  residuals  is  not  due  to  these  planets. 

Noteboom  then  uses  the  twenty  normal  places  of  Witt36  (which  lie 
between  dates  1893  October  31  and  1907  October  8)  and  forms  four 
more  normal  places,  one  in  1910,  one  in  1912,  and  two  in  1914.  He 
then  gets  Elements  AM  out  of  a  least  square  solution  by  a  correc- 
tion of  Elements  AL.  He  gives  the  mass  of  Earth  and  Moon  as 
1/328370±102  and  7r=8".799. 

No  authority  is  available  for  Elements  AK.87 


CELESTIAL   MECHANICS:    LEUSCHNER  49 

REFERENCES 

I  A.  N.    vol.  147,  p.  335.  "A.  N.    vol.  153,  p.  25. 

3  A.  N.  vol.  148,  p.  27.  C.  R.  127,  33A.  N.  vol.  153,  p.  25.  A.  J.  B. 

p.  806.  1900,  p.  158. 

3  A.  J.    vol.  19,  p.  120.  M  A.  N.    vol.  153,  p.  217. 

4 A.  J.  vol.  20,  p.  61.  A.  N.  vol.  "A.  N.  vol.  153,  p.  217.  A.  J.  B. 

148,  p.  143.  A.  J.  B.  1899,  p.  132.  1900,  p.  158. 

6  A.  J.    vol.  19,  p.  148.  35A.  N.    vol.  154,  p.  142. 

6  A.  J.    vol.  19,  p.  155.  *A.  N.    vol.  154,  p.  142.    B.  J.  1903. 

7  A.  J.    vol.  19,  p.  160.    A.  J.  B.  1899,  3TA.  N.    vol.  156,  p.  327. 

p.  132.  *  A.  N.    vol.  156,  p.  328.    B.  J.  1904. 

8  A.  N.    vol.  148,  p.  189.  A.  J.  B.  1901,  p.  184. 

9 A.  N.    vol.  147,  p.  221.  ""A.  N.    vol.   155,   p.  25.    A.  J.   B. 

"A.  J.    vol.  19,  p.  147.  1901. 

II  A.  J.    vol.  20,  p.  8.    A.  J.  B.  1899,  80B.  J.  1905,  p.  534  and  p.  428. 

p.  132.  81G.  WITT;   Untersuchung   iiber  die 

"A.  J.    vol.  20,  p.  134.    A.  J.  B.  1899,  Bewegung    des    Planeten    (433)    Eros, 

p.  132.  Berlin,  1905,  Druck  der  Norddeutschen 

13  A.  J.    vol.  21,  p.  25.  Buchdruckerei.    A.  N.    vol.  176,  p.  211- 

14  M.  N.    vol.  60,  p.  614.    A.  J.  B.  213. 

1900,  p.  158.  33B.  J.    1908,  p.  496 

15  A.  N.    vol.  150,  p.  362.    A.  J.  B.  83B.  J.    1909. 
1899,  p.  132.  34  B.  J.    1916. 

"A.  N.    vol.  147,  p.  363.  85A.  N.    vol.  214,  p.  153. 

"A.  N.    vol.  147,  p.  397.    B.  J.  1901.  3<l"Uber  die  Notwendigkeit  einer  Ver- 

18  A.  N.    vol.  148,  p.  271.    A.  J.  B.  besserung  der  Masse  des  Systems  Erde- 

1899,  p.  132.  Mond,"  by  G.  Witt.    Vierteljahrsschrift 

18  A.  N.  vol.  151,  p.  130.  der  A.  G.  1908,  p.  295. 

20  A.  N.  vol.  151,  p.  137.  A.  J.  B.  S7B.  J.  1918,  p.  (12). 
1899,  p.  132.  B.  J.  1902. 


50 


CELESTIAL   MECHANICS:   LEVSCHNER 


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CELESTIAL   MECHANICS:    LEUSCHNER  51 

(447)  VALENTINE   (1899  ES). 

Discovered  by  Wolf  and  Schwassmann  1  at  Heidelberg  1899,  Octo- 
ber 27. 

Preliminary  elements  were  computed  by  Kreutz2  based  on  observa- 
tions 1899,  October  29,  November  11  and  December  3.  Elements  A. 

Corrections  to  the  ephemendes  based  on  improved  elements  in  19023 
Aa  +16S,  AS  —  2';  in  190£*  January  13,  Aa  +36S,  AS—  0'.3. 

A  more  complete  investigation  of  the  orbit  was  undertaken  by  Hans 
Osten.5  He  received  from  Kreutz  five  sets  of  elements,  B  to  F,  which 
refer  to  different  epochs  in  order  to  show  the  effects  of  the  perturba- 
tions. Kreutz  states  that  Elements  B  to  E  are  comparable  and  are 
to  be  preferred  to  Elements  F.  Elements  B  to  E  were  determined  from 
five  normal  places  (1899  to  1904).  The  residuals  for  the  normal 
places  are 

la  Ib  II  III  IV  V 

AacosS        +5'1         -f-8'7         +i:2         -3'2         -0'7         +35^5 
A5  +14        +8-1         -1.1         +4-0         -0-2         -  3.0 

For  the  purpose  of  investigating  the  general  perturbations  for  this 
planet,  Osten  makes  use  of  Kreutz'  Elements  B,  Leverrier's  elements 
for  Jupiter  and  Saturn,  except  for  the  adoption  ofTETilFs  values  of  the 
mean  motion  for  each,  for  Jupiter's  mass  Newcomb's  value  and  for 
Saturn's  mass  Bessel's  value.  The  perturbations  of  the  first  order  of 
the  masses  are  determined  according  to  Hansen's  method.  As  a  test 
on  his  results,  he  compares  observations  with  theory  for  seven  normal 
places  (1894  to  1906)  with  the  following  results: 

Aa  cos  6  A6 

1.  1894  -10!73*  -  0;54* 

2.  1899a  +5.07  +  1.41 

3.  1899b  +  8.28  +  8.06 

4.  1901  +  0.32  —  1.01 

5.  1902  +  2.44  +  1.39 

6.  1904  +33.26  +16.12 

7.  1906  +32.06  —  2.52 

On  the  basis  of  these  residuals  the  elements  were  corrected,  which 
resulted  in  Elements  G.  The  comparison  between  observation  and 
computation  for  the  normal  places  gives: 

*  Weight  1/9.    Observation  uncertain. 


52  CELESTIAL   MECHANICS:    LEVSCHNER 

Aa  cos  6  A5 

1.  1894  +8!76t  +7!22 

2.  1899a  —2.81  —  2.36 

3.  1899b  +1.57  +4.86 

4.  1901  —2.51  +0.73 

5.  1902  +1.86  +2.00 

6.  1904  —0.68  —0.31 

7.  1906  +1.97  —1.05 

The  approximate  perturbations  due  to  Mars  were  found  to  be  in- 
effective since  they  remained  less  than  0".01. 

For  the  purpose  of  improving  Elements  G  by  means  of  additional 
observations,  Osten6  first  computes  special  perturbations  due  to  Jupiter 
and  Saturn  by  the  method  of  variation  of  elements,  and  determines 
new  Mean  Elements  H  from  the  corrections  which  the  observations 
successively  indicate.  For  the  other  six  major  planets  the  general 
perturbations  are  computed  after  the  method  in  Tisserand,  Vol.  1, 
Chap.  22.  All  the  masses  are  taken  from  Bauschinger,  "Tafeln  z. 
Theor.  Astr."  From  this  combination  of  special  and  general  pertur- 
bations a  set  of  osculating  elements  for  the  ten  oppositions  1899-1912 
results,  forming  the  basis  for  the  formation  of  normal  places.  A  de- 
tailed investigation  of  the  comparison  stars  and  the  observations  leads 
to  normal  places  with  relative  weights  and  corrections  for  magnitude 
equation  in  a.  The  equations  of  conditions  are  solved  under  four 
different  assumptions,  thus  Elements  I  without  and  Elements  J  with 
magnitude  equation.  No  definite  solution  is  accepted,  but  the  impor- 
tance of  using  a  observations  free  from  magnitude  equation  is  em- 
phasized. 

In  the  next  work  Osten7  proceeds  to  determine  the  perturbations 
of  the  second  order  starting  with  Elements  I.  Hansen's  method  is 
followed  throughout  (correcting  an  error  in  "Auseinandersetzung  II, 
page  98,"  which  acts  nearly  as  a  change  of  the  perturbing  mass).  A 
capital  difficulty  is  encountered  in  the  near  commensurabilities  (+2e 
— 5e'  +le")  with  a  period  of  8650  years. 

Osten  proposes  to  treat  such  inequalities  by  expansion  into  power 
series  in  the  time  and  then  eliminate  one  of  the  anomalies.  For  (447) 
the  term  3e  — 7e'  is  thereby  much  enlarged.  A  comparison  is  made 
between  the  special  and  general  perturbations  by  Jupiter  and  Saturn. 
Some  deviations  of  the  order  of  V  are  attributed  to  the  perturbations 
of  the  third  order.  Eight  tables  contain  the  perturbations. 

In  the  hope  of  obtaining  as  accurate  results  for  (447)  as  for  the 
major  planets  Osten8  completes  his  former  theory  and  gives  the  last 

t  Uncertain. 

j  Normal  places  from  Kreutz. 


CELESTIAL   MECHANICS:   LEUSCHNER 


part  of  the  perturbations  of  the  second  order  and  those  of  the  third  in 
longitude  and  radius  vector.  The  accuracy  of  V  in  100  years  is 
extremely  difficult  to  obtain.  A  first  test  is  the  comparison  between 
special  and  general  perturbations  in  the  longitude  and  radius  vector. 
He  finds  small  deviations  which  probably  are  due  to  the  computation 
of  the  special  perturbations. 

As  a  further  test  Osten  gives  the  comparison  with  16  normal  places 
1894-1918.  The  Jupiter  mass  is  also  included  as  an  unknown.  The 
value  1  :  1047.49  is  found.  1  :  1047.35  according  to  Newcomb  is  adopted. 
Thus  Elements  K  are  found.  The  representation  of  the  normals  from 
micrometer  observations  is  —  V  to  +2"  in  the  plane. 


XA.  N.  vol.  150,  p.  431. 

2  A.  N.  vol.  151,  p.  159. 

8  A.  N.  vol.  158,  p.  271. 

4  A.  N.  vol.  170,  p.  195. 


REFERENCES 

5  Astronomische    Abhandlungen    No. 
15,  1908. 
•A.  N.    vol.  194,  No.  4639,  p.  113. 

7  A.  N.    vol.  199,  p.  393. 

8  A.  N.    vol.  210,  p.  129. 


TABLE  13.— Elements— (447)  Valentine  (1899  ES) 


Letter 

Epoch 

M.  T. 

M. 

M 

Q 

i 

A  

1899  Dec.     3.5  

Berlin  

4  21  32 

319  16  21 

72  18  33 

4  49  33 

B  

c  

1899  Dec.     5.5  
Osc.  Nov.     5.0  
1901  Feb.     8.0  

Berlin  
Berlin  

4  40  42 
87    2  32 

319  15     3 
318  53  31 

72  24    5 
72  24    0 

4  49     4 
4  49     4 

D 

1902  Apr      4  0 

Berlin 

167  42  14 

318  29  20 

72  23  32 

4  49     4 

E  

1904  Oct.     10.0  

Berlin  

345  41  51 

316  22  10 

72  19  44 

4  49     5 

F 

1906  Feb       2  0 

Berlin 

79  55  40 

313  38  14 

72  25    5 

4  49  21 

G 

1899  Nov      5  0 

Berlin 

358  57  25 

319  13  45 

72  24  10 

4  49     4 

H 

1899  Nov.     5  0 

Berlin 

2  42  27 

315  27  43 

72  25  38 

4  49     9 

1 

1899  Nov.     5  0 

Berlin 

358  52  18 

319  13  42 

72  24  16 

4  49     4 

j 

1899  Nov.     5  0 

Berlin 

358  52  21 

319  13  42 

72  24  11 

4  49     3 

K 

1899  Nov.     5  0 

Berlin 

358  57  21 

319  13  40 

72  24  17 

4  49     4 

Letter 

Epoch 

M.  T. 

*> 

ft 

Equinox 

Authority 

A 

1899  Dec.     3.5  

Berlin  

0       /            * 

2  36  37 

* 
687.012 

1900.0 

H.  Kreutz 

B   .    ... 

1899  Dec.     5.5  

Berlin  

2  34  33 

687.5969 

1900.0 

H.  Kreutz 

c 

Osc.  Nov.     5.0  
1901  Feb       8  0 

Berlin 

2  35    5 

687.6846 

1900.0 

H.  Kreutz 

D 

1902  Apr      4  0 

Berlin 

2  36  17 

688.1604 

1900.0 

H.  Kreutz 

E 

1904  Oct     10  0 

Berlin    

2  40  15 

686.5435 

1900.0 

H.  Kreutz 

F 

1906  Feb       2  0 

Berlin  

2  38  34 

687.9066 

1910.0 

H.  Kreutz 

G   . 

1899  Nov.     5  0 

Berlin  

2  34  32 

687.3550 

1900.0 

Osten* 

H  

1899  Nov.     5  0 

Berlin  

2  25  42 

687.3504 

1900.0 

Osten 

I  

1899  Nov.     5  0 

Berlin  

2  34  32 

687.6016 

1900.0 

Osten 

J  

1899  Nov.     5.0 

Berlin  

2  34  32 

687.6018 

1900.0 

Osten 

K  

1899  Nov.     5.O..    .. 

Berlin  

2  34  32 

687.3884 

1900.0 

Osten 

Mean  elements. 


54  CELESTIAL   MECHANICS:   LEUSCHNER 

(588)  ACHILLES,  1906  TG. 

Discovered  by  Wolff1  at  Heidelberg  1906,  February  22. 

From  observations  of  February  22,  and  March  5,  Berberich2  com- 
puted Elements  A  for  a  circular  orbit  (search  ephemeris  for  April 
1906). 

The  first  elliptic  Elements  B  were  published  by  Berberich.3  They 
are  based  on  observations  1906,  February  22,  March  23,  and  April 
22.  An  ephemeris  for  May  and  June,  1906,  is  also  included.  He  also 
points  out  that  the  aphelion  point  lies  far  beyond  Jupiter's  orbit, 
and  that  the  present  orbit  has  had  its  form  and  position  for  a  long 
time. 

From  Elements  B,  Charlier4  finds  that  Achilles  is  approximately 
55°  ahead  of  Jupiter,  consequently  very  close  to  one  of  Lagrange's 
libration  points.  He  also  points  out  that  we  may  have  here  a  case 
(as  he  has  shown)5  where  the  planet  does  not  remain  at  the  apex  of 
the  equilateral  triangle,  as  suggested  by  Lagrange,  but  oscillates  about 
it  with  a  period  of  about  148  years. 

Comments  on  the  character  of  the  orbit  of  Achilles  have  been  pub- 
lished by  Berberich,6  Crommelin,7  Ristenpart,8  Stroobant.9 

On  the  basis  of  observations  extending  from  1906,  February  22,  to 
May  19,  Bidschof10  has  published  a  set  of  Elements  C  and  an 
ephemeris  for  1907.  A  continuation  of  the  ephemeris  with  corrections 
to  the  same  was  published  by  Bidschof.11  An  observation  1907, 
February  12,  gives  Aa  — 51s,  and  AS  +6'.7. 

An  ephemeris  for  1909  based  on  Elements  B.  J.  1911  with  special 
perturbations  due  to  Jupiter  was  published  by  Franz.12 

For  the  following  years,  to  1919,  the  B.  J.  (Kleine  Planeten)  pub- 
lishes elements  by  Bidschof  brought  forward  to  a  new  epoch  and 
mean  equinox,13  Elements  D. 

As  an  application  of  Leuschner's14  satellite  method,  Einarsson15 
computed  a  preliminary  orbit,  (Elements  E),  based  on  observations 
1907,  February  12,  April  15,  and  June  2.  Special  perturbations,  due 
to  Jupiter,  computed  by  Encke's  method,  were  included  for  the  period 
covering  the  observations.  The  maximum  residuals,  for  nine  observa- 
tions of  1907,  were  Aa  cos  8  — 3".6,  AS  ±0".7.  Elements  E  were 
used,  without  perturbations,  to  represent  an  observation  1906,  May 
19,  with  the  following  results:  Aa  cos  8  +1'01".8,  AS  —  39".2. 

The  most  recent  work  on  Achilles  was  done  by  Julie  M.  Vinter- 
Hansen.16  With  Bidschof's  Elements  D,  the  special  perturbations, 
due  to  Jupiter  and  Saturn,  were  computed  from  1906  to  1914.  All 
observations  of  1906  and  1907,  two  of  1913,  and  one  of  1914,  were 
then  represented  and  residuals  determined.  With  these  as  a  basis, 


CELESTIAL   MECHANICS:   LEUSCHNER 


55 


eleven  normal  places  were  formed,  weighted  according  to  number  of 
observations  in  each. 

The  residuals,  (freed  from  perturbations),  for  the  1906  and  1907 
normal  places  are  small.  For  normal  place  X,  (1913),  AacosS 
+2781".2,  AS  +  1685".0;  for  normal  place  XI,  (1914),  Aa  cos  8 
+1664".!,  A8+852".7.  The  resulting  orbit  improvement  gave  Ele- 
ments F.  The  residuals  for  the  normal  places  from  Elements  F  vary 
from  +  19".8  to  —  40".9  in  Aa  cos  8,  and  —  9".2  to  +23"8  in  A8. 
With  Elements  F  the  special  perturbations  were  recomputed  by  J. 
Braae.  With  Elements  F  and  the  new  perturbations,  new  residuals 
for  the  normal  places  were  determined  and  the  orbit  improved  on  this 
basis.  When  the  corrections  to  the  elements  were  put  back  into  the 
equations  of  condition,  the  residuals  were  of  the  same  order  as  prior  to 
the  solution.  Miss  Hansen  concludes  a  normal  place  must  be  incor- 
rect. The  equations  of  condition  were  again  solved,  with  the  last 
normal  place  omitted.  The  resulting  Elements  G  represent  the  ten 
normal  places  in  Aa  cos  8  — 1".5  to  +5".8  and  A8  — 1".3  to  +2".3. 

The  normal  place  XI  gives  residuals  Aa  cos  8  — 1817".2,  A8 
— 1172".0,  which  indicates  that  it  does  not  belong  to  Achilles. 

The  special  perturbations  due  to  Jupiter  for  1915  to  1920  are  pub- 
lished by  Miss  Hansen.17  They  were  computed  from  Elements  G. 
These  elements  are  published  and  utilized  in  Kleine  Planeten  since 
1920.  In  Ark.  for  Math.  Bd.  4  Nr.  20  Linders  developed  the  approxi- 
mate theory  for  planets  near  the  libration  points  and  gives  the  prin- 
cipal perturbations  of  the  elements  of  588.  Cf.  Heinrich  V.  J.  S.  1913. 


REFERENCES 


1  A.  N.    vol.  170,  p.  353. 

2  A.  N.    vol.  171,  p.  11. 


8  A.  N.    vol.  171,  p.  127. 

4  A.  N.    vol.  171,  p.  213. 

5  Meddelanden  fran  Lunds  Observa- 
torium,  No.  18. 

•Nat.  Rund.    vol.  21,  p.  485^86. 
7  Observatory,    vol.    29,    p.    352-355. 
Pop.  Ast.    vol.  14,  p.  472-475. 
8H.  u.  E.    vol.  18,  p.  517-521. 


•Ciel  et  Terre.    vol.  27,  p.  161-164. 

10  A.  N.    vol.  174,  p.  45-48. 

11  A.  N.    vol.  174,  p.  175. 
"A.  N.    vol.  180,  p.  295. 

13  B.  J.    1914,  p.  30. 

14  L.  O.  Pub.    vol.  vii,  p.  455. 
15 In  Manuscript  (Berkeley). 

18  Pub.    og    mindre    Meddelelser    fra 
Kobenhavns  Obs.    No.  37. 
17  A.  N.    vol.  208,  p.  345. 


TABLE  U.— Elements— (688)  Achilles  (1906  TG.) 


Letter 

Epoch 

M.  T. 

M. 

• 

Q 

i 

A 

1906  Mar     5  5 

Berlin 

(u)  186°  15  '9 

316°  1/2 

11°  43/2 

B 

1906  Feb  22  5 

Berlin 

0         /            » 

48  57  24 

o       /        • 

120  25  50 

o       /        • 
315  34     7 

Of           • 

10  20  53 

C  

1906  Feb.  22.5  

Berlin  

43  45  37 

(129  24  11 

315  31     7 

10  16  36 

D 

1907  Apr  15  5 

Berlin    .  . 

80  18  12 

\129  24  10 
125  37  50 

315  31  58 
315  36     2 

10  16  36 
10  18  25 

E  
F  
G  

1907  Apr.  15.66... 
1907  May  28.0.... 
1907  May  28.0  

Greenwich.  . 
Berlin  
Berlin  

80  23  50 
82  54  47 
84     3     2 

125  25  21 
127     7  10 
125  36  22 

315  35  45 
315  34  26 
315  35  59 

10  18  45 
10  17  53 
10  18  14 

56  CELESTIAL  MECHANICS:   LEUSCHNER 

TABLE  U—Ekments—(588)  Achilles  (1906  TG.)-  Continued 


Letter 

Epoch 

M.  T. 

<f 

n 

Equinox 

Authority 

A 

1906  Mar     5  5 

Berlin 

o    /     // 

• 

312  32 

1906  0 

Berberich 

B 

1906  Feb  22  5 

Berlin 

9  38  43 

295.133 

1906  0 

Berberich 

c 

1906  Feb  22  5 

Berlin 

8  10  15 

294  .  703 

[  1906.0 

Bidschof 

D 

1907  Apr  15  5 

Berlin     .  . 

8  42  54 

295  464 

\1907.0 
1910.0 

Bidschof 

E 

1907  Apr  15  66 

Greenwich 

8  45  49 

295.6847 

1907.0 

Einarsson 

F 

1907  May  28  0    . 

Berlin.   .. 

8  25  19 

294.71497 

1910.0 

Miss  Hansen 

G 

1907  May  28  0     . 

Berlin  ... 

8  36  13 

295  96333 

1910.0 

Miss  Hansen 

CELESTIAL  MECHANICS:   LEUSCHNER  57 

(617)  PATROCLUS,  1906  VY. 

Discovered  by  Kopff1  at  Heidelberg,  October  17,  1906. 

Preliminary  Elements  A  were  computed  by  Heinrich2  based  on 
four  observations  October  21  to  December  7. 

From  Elements  A  Charlier3  has  noted  that  the  longitude  of  Pat- 
roclus  is  approximately  60°  behind  Jupiter  as  compared  with  Achilles 
and  Hector  which  are  approximately  60°  ahead  of  Jupiter.  From 
the  same  elements,  Stromgren3  has  made  a  similar  comparison. 

A  second  set  of  preliminary  Elements  B  were  computed  by  Hein- 
rich4 by  variation  of  the  distances  on  the  basis  of  five  observa- 
tions from  1906  October  21  to  December  7.  On  page  339  of  the 
same  reference  the  correction  is  made  that  Elements  B  refer  to 
equinox  1906.  Elements  B  are  used  to  compute  the  special  per- 
turbations, by  the  method  of  variation  of  the  constants,  and  a  third 
set  of  Elements  C  with  an  ephemeris  for  1907  is  published.5  A 
continuation  of  the  ephemeris  is  published  on  page  251  of  the  same 
reference.  Observations  November  8  and  10,  1907,  show  the  follow- 
ing corrections  to  the  ephemeris:  Aa= — 34s  A8= — 0'.8. 

Ephemerides  for  the  oppositions  1908,  1909,  1910,  1912,  1913,  by 
Heinrich6  are  based  on  Elements  C.  The  correction  to  the  ephemeris 
in  1909  was  Aa— 39s  and  A8+3'.3. 

From  1914  to  1918  the  ephemerides  based  on  Elements  C  are 
published  in  Kleine  Planeten. 

An  extensive  application  of  Wilkens'  method7  for  planets  of 
Jupiter's  group,  is  made  by  Drucker.8  He  first  takes  Heinrich's  Ele- 
ments C  as  a  basis  for  the  computation  of  the  perturbations  due  to 
Jupiter  and  Saturn,  by  the  method  of  variation  of  elements.  Then 
corrects  Elements  C  on  the  basis  of  twelve  normal  places  (1906  to 
1918).  See  Elements  D.  The  residuals  for  the  twelve  normal  places, 
from  Elements  D  plus  the  special  perturbations  due  to  Jupiter  and 
Saturn,  vary  from  —  7".9  to  +10".3  for  Aa  cos  8  and  — 4".8  to 
+4".9  for  AS. 

With  Elements  D  the  perturbations  due  to  Jupiter  and  Saturn 
are  again  computed  and  with  the  former  twelve  normal  places  and 
a  new  thirteenth  normal  place,  (1919),  the  Elements  D  were  cor- 
rected to  a  new  set,  E.  With  this  new  set  of  Elements  E,  and  the 
new  perturbations,  the  residuals  for  the  normal  places  vary  from 
— 3".4  to  +3".5  for  Aa  cos  8  and  from  — 2".8  to  +2".9  for  AS.  He 
states  these  Elements  E  may  be  considered  as  definite.  In  a  foot- 
note on  page  27  of  the  same  reference,  the  residuals  for  an  observation 
1920,  December  16,  are  Aa  cos  8  +4".4  and  AS  +1".4.  The  ephemeris 
for  1920  was  computed  from  Elements  E,  plus  special  perturbations 
due  to  Jupiter  and  Saturn. 


58 


CELESTIAL   MECHANICS:   LEUSCHNER 


Then  he  derives  corresponding  elements  in  Wilkens'  system,  Ele- 
ments F  and  G.  Elements  F  correspond  to  Elements  D,  and  Elements 
G  correspond  to  Elements  E.  He  states  that  it  was  only  necessary 
according  to  Wilkens'  method  to  compute  Elements  F. 

He  now  computes  positions  (a  and  8),  for  the  thirteen  normal 
places,  (1)  from  Elements  E,  with  special  perturbations  due  to 
Jupiter;  (2),  the  same  places  from  Elements  E  without  perturba- 
tions; (3),  the  same  places  from  Elements  G  without  perturbations; 
(4),  the  same  places  from  Elements  E  with  special  perturbations  due 
to  Jupiter  and  Saturn.  A  comparison  is  then  made  between  the  posi- 
tions computed  by  the  various  systems,  (1),  (2),  (3),  (4).  The  table 
of  differences  shows  that  Saturn's  perturbations  do  not  become  ef- 
fective until  1918.  It  shows,  as  we  should  expect,  that  for  the  period 
of  osculation,  the  various  systems  give  the  same  results.  It  finally 
shows  that  positions  computed  from  (3)  come  closer  to  those  com- 
puted by  (1),  than  the  places  computed  by  (2). 

Finally  he  computes  the  special  perturbations  due  to  Jupiter,  by 
the  variation  of  elements,  with  the  aid  of  Elements  G  and  shows 
that  they  are  of  the  same  order  as  those  computed  with  Elements  E. 

The  ephemerides  published  in  Kleine  Planeten  for  1920  and  1921 
are  based  on  Elements  E,  plus  special  perturbations  due  to  Saturn 
and  Jupiter.  J^^  ^  ^f( 

An  ephemeris  for  the  opposition  using  Elements Ts^  19l8-1919, 
was  computed  by  Paul  Maitre.9  Ephemerides  for  oppositions  1919- 
1920,  1920-1921,  and  1922,  are  also  published  by  Drucker.10 

REFERENCES 


A.  N.  vol.  172,  p.  387. 

A.  N.  vol.  175,  p.  87. 

A.  N.  vol.  175,  p.  89. 

A.  N.  vol.  175,  p.  291. 

A.  N.  vol.  176,  p.  193. 

A.  N.  vol.  179,  p.  223— vol.  180,  p. 


45— vol.  183,  p.  207— vol.  190,  p.  395— 
vol.  194,  p.  208. 

7  A.  N.  vol.  205,  No.  4906. 

8  A.  N.  vol.  214,  No.  5114. 
9Cir.  O.  M.  No.  85,  1918. 

10B.-Z.  der  A.  N.  No.  14,  1919-No. 
48,  1920— No.  1,  1922. 


TABLE  15.— Elements  (617)— Patroclus  (1906  VY) 


Letter 

Epoch 

M.T. 

M 

0) 

0 

i 

A 

1906  Oct.    21  5  

Berlin.... 

41  31  55 

297  28  37 

43  21  39 

22  16  47 

B.. 

1906  Nov.  29.0  

Berlin  

41  27  30 

302  11  27 

43  25  32 

22     3  33 

c  

1907  Dec.   14.0  

Berlin  

73     1  25 

302  25  48 

43  28  36 

22     3  15 

D  

1906  Nov.  29.0  

Berlin  

42  00  14 

-58  26  46 

43  27  49 

22     7  00 

E  

1906  Nov.  29.0  

Berlin  

41  59  24 

-58  25  39 

43  27  49 

22     6  57 

F 

1906  Nov    29  0 

Berlin 

42  19     5 

-58  45  27 

43  27  49 

22     7  00 

G  

1906  Nov.  29.0  

Berlin  

42  18  15 

-58  44  21 

43  27  49 

22     6  57 

CELESTIAL  MECHANICS:   LEUSCHNER 
TABLE  15— Elements— (617}  Patrodus  (1906  V  F.)— Continued 


59 


Letter 

Epoch 

M.T. 

f 

M 

Equinox 

Authority 

A 

1906  Oct     21  5 

Berlin    . 

o           • 
8  42  41 

• 
300  145 

1906  0 

Heinrich 

B 

1906  Nov    29  0 

Berlin.   . 

8  16     7 

300  659 

1906  0 

Heinrich 

C 

1907  Dec     14  0 

Berlin... 

8  14  38 

300  532 

1910.0 

Heinrich 

D 

1906  Nov    29  0 

Berlin  

8  15  33 

300.7805 

1910  0 

Drucker 

E 

1906  Nov    29  0 

Berlin  

8  15  32 

300.7654 

1910.0 

Drucker 

F 

1906  Nov    29  0 

Berlin  

8  13    9 

301.4459 

1910.0 

Drucker 

G   . 

1906  Nov    29.0       

Berlin  

8  13    9 

301.4309 

1910.0 

Drucker 

60  CELESTIAL   MECHANICS:   LEUSCHNER 

(624)  HECTOR,  1907  XM. 

Discovered  by  Kopff1  at  Heidelberg  on  February  10,  1907,  >)/V  W,63. 

A  preliminary  orbit,  Elements  A,  was  computed  by  Stromgren2  based   Afi 
on  observations  from  February  10  to  April  16.    These  elements  gave 
the  following  residuals: 

1907  AA  A£ 

Feb.      10  — 1".0  +2".2 

April    19  +2  .8  +8  .3 

He  reports  that  Hector  is  another  planet  with  mean  motion  nearly 
equal  to  that  of  Jupiter.  Since  the  perturbations  of  Jupiter  are  small 
and  the  perturbations  of  the  other  planets  will  be  ineffective  for  a 
long  time,  this  planet  will  remain  in  the  neighbourhood  of  the  libration 
point  for  a  long  time. 

An  observation  by  Palisa  February  29,  1908,  gives  a  correction  to 
f7,  Jl3  the  ephemeris3  based  on  Elements  A  as  follows:  Aa=— 37s  A8=+6'.3. 
r 0,317  An  ephemeris  is  published  by  Stromgren4  for  the  opposition  of  1909 
based  on  Elements  A. 

In  preparation  for  the  ephemeris  of  1911,  Stromgren,5  assisted  by 
J.  Fischer-Petersen,  computes  a  new  set  of  Elements  B,  based  on  nine 
normal  places  (oppositions  1907,  1908,  1909),  taking  into  account 
the  perturbations  due  to  Jupiter  and  Saturn.  These  elements  represent 
the  normal  places,  AacosS  between  +0'.30  and  — .17,  and  AS  between 
— 0'.04  and  +OM6. 

The  planet  was  next  observed  in  July  1911.  The  comparison  be- 
tween observation  and  ephemeris  (from  Elements  B)  was  unsatisfac- 
tory.  Stromgren,  assisted  by  Ruben  Andersen,6  again  investigated  the 
orbit  based  on  observations  from  1907  to  1911.  For  this  purpose 
Elements  A  were  used  to  compute  the  residuals  for  five  observations, 
July  4  to  16,  1911.  A  tenth  normal  place  was  formed  from  these  five 
places  with  the  following  residuals  Aacos8  =  —52'  23" .9  AS  =  —11' 
36". 6.  A  weight  of  five  (number  of  observations)  was  given  to  this 
normal  place.  Combining  this  normal  place  with  the  solution  that 
led  to  Elements  B,  a  new  set  of  elements  was  derived,  Elements  C. 
With  these  elements  the  special  perturbations,  due  to  Jupiter  and 
Saturn,  were  computed  by  the  method  of  Encke,  for  1906  to  1912. 
The  observations  for  1907  and  1908  were  then  computed  without  taking 
into  account  the  perturbations  (reported  as  being  small)  and  the  obser- 
vations of  1909  and  1911  were  represented  with  perturbations.  From 
these  representations  ten  new  normal  places  were  formed  and  new 
Elements  D  were  obtained  from  a  least  square  solution.  This  set 


CELESTIAL   MECHANICS:    LEUSCHNER  61 

represents  the  normal  places,  AaCOsS  between  — 7".l  and  +3".8,  and 
AS  between  — 1".6  and  +3".0. 

With  Elements  D  and  special  perturbations,  the  maximum  residuals 
for  opposition  of  1912  were  Aacos8  =  +2M2  and  A8  =  +26".6. 
Stromgren,  assisted  by  Julie  M.  Vinter  Hansen,7  again  investigated  the 
orbit  of  Hektor  on  the  basis  of  thirteen  normal  places  (1907  to  1912). 
A  least  square  solution  led  to  Elements  E.  At  the  conclusion  of  this 
work,7  osculating  elements  were  computed  for  the  epoch  of  each  year, 
1907  to  1913,  based  on  Elements  E  and  the  special  perturbations  pub- 
lished in  the  same  reference.  The  ephemeris  for  1913  was  published 
in  A.  N.,  vol.  198,  p.  367. 

In  A.  N.,  vol.  200,  pp.  79-82,  Stromgren  publishes  the  comparisons 
between  the  observations  and  the  ephemeris,  which  was  based  on 
Elements  E  and  the  perturbations  due  to  Jupiter  and  Saturn,  for  1913 
and  1914.  The  maximum  residual  for  1913  for  Aacos8=+08.53  and  for 
A8=+7".l.  For  1914  the  maximum  residual  for  AacosS=+03.51 
and  AS=+H"-6-  Stromgren  regards  these  residuals  as  satisfactory. 

As  an  application  of  Leuschner's  satellite  method,  S.  Einarsson3 
computed  a  preliminary  orbit  of  Hector  based  on  observations  1907, 
February  10,  March  11,  and  April  16,  Elements  F.  These  elements 
represented  an  observation  of  May  2,  1908,  as  follows:  Aacos8  = 
—4'  32".4  A8=+3'  31".9.  With  Elements  F,  the  special  perturba- 
tions by  Encke's  method  were  computed  for  the  1907  opposition.  New 
elements  were  then  computed  from  observations  1907,  February  10, 
March  11,  and  a  normal  place  from  observations  April  12,  16,  and  19, 
by  a  differential  correction  of  Elements  F.  These  new  Elements  G 
represented  the  opposition  of  1908,  May  2,  as  follows:  Aa  +  30",  AS  -^. 
95",  and  for  1909,  April  17,  Aa  +  6M;  AS  — 6'.7.  The  application  of 
Leuschner's  method  thus  yielded  far  better  results  than  the  ordinary 
method  followed  by  Stromgren:  1908,  Aa=—  9' 15"  AS=+6'  18". 
No  further  work  was  done  on  this  planet  by  Einarsson  since  Strom- 
gren and  his  colleagues  had  already  made  extensive  investigations. 

An  ephemeris  for  1915  is  published  by  J.  Fischer-Petersen,9  using     AN. 
Elements  E  and  taking  into  account  perturbations  due  to  Jupiter  and 
Saturn. 

The  elements  and  ephemeris  for  1916 10  are  based  on  Stromgren's    KPJ; 
Elements  E  with  special  perturbations  by  Jupiter  and  Saturn  brought 
forward. 

Elements  and  ephemerides  are  given  in  Kleine  Planeten  for  each 
year  to  1921.  No  further  comment  is  published  regarding  the  elements. 

An  ephemeris  is  published  by  M.  Henri  Blondel11  for  the  opposition      M« 
of  1921  which  has  been  corrected  on  the  basis  of  an  observation  at 


62 


CELESTIAL   MECHANICS:   LEUSCHNER 


/  J7  /  Algiers12  on  May  6,  1921.  This  observation  gave  a  correction  to  the 
ephemeris  published  in  Kleine  Planeten  of  +7m-0  in  right  ascension 
and  — 80'  in  declination. 

In  A.  N.,  vol.  215,  p.  249,  A.  Wilkens  has  published  an  article,  "Uber 
die  Sakularen  Veranderungen  der  Grossen  Achsen  der  Bahnen  der 
Planeten  der  Jupiter  Gruppe." 

In  A.  N.,  vol.  175,  p.  89,  Charlier  gives  a  brief  discussion  on  the 
orbits  of  the  Trojan  group,  regarding  their  motion  about  the  libration 
points. 

In  A.  N.,  vol.  206,  p.  235,  A.  Koref  outlines  his  investigation  regard- 
ing the  motion  of  Hector.  His  preliminary  work  is  based  on  eighteen 
normal  places  (1907  to  1914).  The  investigation  will  be  completed 
when  observations  of  1918  and  1919  are  available. 

REFERENCES 


'A.  N.    vol.  174,  p.  63. 

aA.  N.    vol.  175,  p.  14. 

'  A.  N.    vol.  177,  p.  123. 

4  A.  N.    vol.  180,  p.  327. 

6  Publikationer  og  mindre  Meddel- 
elser  fra  Kobenhavns  Observatorium 
No.  6.  A.  N.  vol.  188,  p.  395. 

*  Publikationer  og  mindre  Meddel- 
elser  fra  Kobenhavns  Observatorium 
No.  8. 


7  Publikationer  og  mindre  Meddel- 
elser  fra  Kobenhavns  Observatorium 
No.  12. 

8 In  manuscript  (Berkeley). 

9  A.  N.  vol.  201,  p.  335.  B.  A.  J. 
1917. 

10Eph.  der  Kleinen  Planeten,  1916, 
p.  17,  p.  77,  p.  93,  p.  97. 

"Cir.  O.  M.    No.  480  (1921). 

"Cir.  0.  M.    No.  171,  second  series. 


TABLE  16.— Elements— (624)  Hector  (1907  XM) 


Letter 

Epoch 

M.T. 

M 

M 

0 

i 

A  

1907  Feb.    10.0  

Berlin  

335  47  12 

183  51  52 

341  58  25 

18  7  17 

B  

1907  Feb.    10.0  

Berlin  

343  51  43 

175    6  42 

341  58  57 

18  8  34 

c 

1907  Feb     10  0 

Berlin 

345  38  38 

173     5  26 

341  59  47 

18  9  13 

D  
E  
F  

1907  Feb     10.0  
1907  Feb.    10.0  
1907  Mar.  11.36  

Berlin  
Berlin  
Greenwich  .  . 

343  40  12 
343  48  55 
341  56  43 

175  19    0 
175    9  30 
179  46  25 

341  59  18 
341  59  15 
341  57  32 

18  8  50 
18  8  45 
18  8  05 

G  

1907  Mar.  11.36  

Greenwich  .  . 

348  27  23 

172  44  30 

341  57  14 

18  8  19 

Letter 

Epoch 

M.T. 

<P 

M 

Equinox 

Authority 

A 

1907  Feb     10  0 

Berlin 

Off 

2     8  24 

292  5842. 

1907  0 

Strdmgren 

B 

1907  Feb     10  0 

Berlin 

56  46 

293  1585 

1910  0 

Stromgren 

C 

1907  Feb    10  0 

Berlin 

1  43  45 

295  3661 

1910  0 

Stromgren 

D  

1907  Feb    10  0 

Berlin 

1  56  52 

293.1072 

1910  0 

Stromgren 

E  

1907  Feb.    10  0 

Berlin  ... 

1  56  29 

293  .  1782 

1910.0 

Stromgren 

F  

1907  Mar.  11.36     

Greenwich.  . 

2  03  07 

292.7487 

1910.0 

E  inarsaon 

G  

1907  Mar.  11.36  

Greenwich.  . 

1  55  32 

293.1164 

1910.0 

Einarsson 

CELESTIAL   MECHANICS:    LEUSCHNER  63 

(659)  NESTOR,  1908  CS. 

Discovered  by  Wolf1  at  Heidelberg,  1908,  March  23. 

From  observations  of  1908,  March  25  and  May  2,  Ebell2  computed 
a  circular  orbit.  From  these  Elements  A  it  is  evident  that  the  planet 
belongs  to  the  Trojan  group,  Achilles  type,  and  he  notes  the  planet's 
position  is  near  a  libration  point  (60°  ahead  of  Jupiter). 

Preliminary  elliptic  Elements  B,  with  an  ephemeris  for  1908,  were- 
computed  by  Ebell3  from  observations  1908,  March  23,  April  26, 
and  May  19.  With  Elements  B  Ebell4  publishes  an  ephemeris  for  1909. 

The  improvement  of  Elements  B  was  undertaken  by  Anderson.5 
Special  perturbations  due  to  Jupiter  and  Saturn  were  computed  with 
Elements  B.  Six  normal  places  were  formed  from  eleven  observa- 
tions extending  from  1908,  March  23  to  1912,  September  9. 

The  coefficients  for  the  differential  equations  were  computed  by  the 
method  given  in  Oppolzer.6  The  solution  of  the  equations  gave  ab- 
normal corrections  when  both  or  either  normal  places  IV  and  V  were 
used.  (IV  and  V  are  single  observations).  His  final  solution  was 
based  on  the  first  four  normal  places,  (1908  to  1909).  The  resulting 
Elements  C  were  used  to  compute  the  special  perturbations  due  to 
Jupiter  and  Saturn  and  an  ephemeris  for  1913. 

Wolf  reports  7  that  Nestor  cannot  be  found  at  ephemeris  position. 

An  ephemeris  for  1914  based  on  Elements  C,  plus  perturbations, 
(disturbing  planets  not  stated),  was  published  by  Andersen.8  He 
states  no  other  observations  for  this  planet  are  available.  Reported 
observations  since  1909  do  not  appear  to  belong  to  Nestor.9 

An  ephemeris  for  1915  based  on  Elements  C  by  Anderson,  was 
published  by  Stromgren.10  He  states  the  correction  to  ephemeris, 
based  on  an  observation  of  1914,  December  20,  is  Aa  +62S,  AS  — 7'. 6. 

Another  attempt  to  improve  the  orbit  of  Nestor  was  made  by  An- 
dersen11 in  1917.  In  this  attempt  six  normal  places  were  formed  from 
observations  in  1908,  1909,  and  1917.  The  perturbations  due  to 
Jupiter  and  Saturn  were  computed  from  Elements  C,  and  the 
normal  places  represented  from  the  same  elements.  The  starting 
residuals  for  observation,  1908  and  1909,  were  small,  (maximum 
— 2".4) ;  for  the  normal  place  V,  (two  observations  of  1917),  Aa  cos 
8  — 2008".8,  AS  +458".4;  for  the  normal  place  VI,  (two  observa- 
tions of  1917),  Aa  — 1934".7,  AS  +392".2.  The  differential  correc- 
tions were  computed  as  in  the  earlier  work.6  For  resulting  elements 
see  D.  These  elements  left  the  following  residuals  for  the  V  and  VI 
normal  places: 

Aa  cos  S  AS 

V  —  4".7  —  6".2 

VI  +5.0  +7.2 


64 


CELESTIAL   MECHANICS:    LEUSCHNER 


Elements  D  are  sufficiently  accurate  to  show  that  observations 
reported  for  this  planet  in  1914-1915  do  not  belong  to  Nestor. 

With  Elements  D  and  perturbations  computed  from  Elements  C 
an  ephemeris  was  computed  for  1918. 

Julie  M.  Vinter-Hansen  has  published12  the  special  perturbations, 
(disturbing  planets  not  stated),  computed  by  Pedersen  from  1907  to 
1918.  Also  Andersen's  Elements  D  brought  up  to  epoch  of  1918, 
(Elements  E),  and  an  ephemeris  for  1919. 

Seagrave  has  published13  an  ephemeris  for  1920.  No  reference  is 
made  as  to  what  elements  were  used. 

The  most  recent  work  on  Nestor  is  by  Kristensen.14  He  begins 
with  Andersen's  Elements  D,  and  computes  the  special  perturba- 
tions due  to  Jupiter  and  Saturn  and  then  represents  all  the  observa- 
tions available  from  1908  to  1919.  The  maximum  initial  residuals 
(perturbations  included),  for  the  1919  observations  are  Aa  cos  8 
— 64S.29,  AS  -|-494".8.  He  then  forms  eleven  normal  places  based  on 
all  the  available  observations.  The  maximum  initial  residuals  appear 
in  the  IX  Normal  place,  Aa  cos  8  — 963".0,  AS  +494".0,  (pertur- 
bations included).  The  resulting  solution  of  the  eleven  equations  gives 
Elements  F.  These  elements  represent  the  normal  places  satis- 
factorily. The  maximum  residuals  are  Aa  cos  8  +0S.16,  AS  +7".2. 
He  states  these  Elements  F  may  be  considered  as  a  definitive  system. 

The  elements  published  in  Kleine  Planeten,  1917  to  1920,  are  those 
of  Andersen  (Elements  C),  brought  up  to  new  epochs. 


REFERENCES 


*A.  N.  vol.  177,  p.  287. 

3  A.  N.  vol.  177,  p.  399. 
8  A.  N.  vol.  178,  p.  71. 

4  A.  N.  vol.  180,  p.  213. 

5  A.  N.  vol.  195,  No.  4678,  p.  433. 
*  Oppolzer.    vol.  ii,  p.  390-391. 
TA.  N.  vol.  196,  p.  14. 

8  A.  N.  vol.  199,  p.  221. 


•Veroff.  R.  I.  No.  42,  remarks  5 
and  15. 

10  A.  N.    vol.  200,  p.  56. 

"A.  N.    vol.  206,  No.  4923,  p.  17. 

"A.  N.    vol.  208,  p.  15-16. 

13  A.  J.    vol.  32,  p.  167. 

"Pub.  og  mindre  Meddelelser  fra 
Kobenhavns  Obs.  No.  37. 


TABLE  17.— Elements  (659)— Nestor  (1908  CS.) 


Letter 

Epoch 

M.  T. 

M. 

CO 

D 

i 

A  

1908  Mar.  25.5... 

Berlin  

(u)  196°  55'.8 

350°  55'.  7 

4°40'.l 

B  

1908  Mar.  23.5..  . 

Berlin  

O          1           * 

240  38     5 

327  31  28 

349  57  42 

4  31  15 

C  

1908  Mar.  23.51 

Osculation..  . 

1908  Apr.  12.0  /" 

Berlin  

240     3  56 

328    4  54 

350    0     1 

4  31  31 

D  
Osculation.  .  . 

1908  Mar.  23.  5  1 
1908  Apr.  12.0  J" 

Berlin  

237  29  22 

329  41     1 

350    2    4 

4  31  47 

E  

1918  Mar.  11.0.  .  . 

Greenwich.  . 

179  57  36 

331  58  33 

350    5  22 

4  31  45 

F  

1908  Mar.  23  .  5\  .  . 

Osculation  .  .  . 

1908  Apr.  12.0  J.  . 

Berlin  

238  18  00 

329     7  41 

350     1  29 

4  31  44 

CELESTIAL   MECHANICS:    LEUSCHNER  65 

TABLE  17.— Elements  (659}— Nestor  (1908  CS.)— Continued 


Letter 

Epoch 

M.  T. 

, 

M 

Equinox 

Authority 

A..    .. 

1908  Mar.  25  5 

Berlin 

0        ,            „ 

290  67 

1908  0 

Ebell 

B  

1908  Mar.  23.5. 

Berlin 

6  23  59 

300  785 

1908  0 

Ebell 

c  

1908  Mar.  23.51 

Osculation..  . 
D  

1908  Apr.  12.0  r  ' 
1908  Mar.  23.  5^ 

Berlin  

6  26  44 

301.0002 

1910.0 

R.  Andersen 

Osculation  .  .  . 
E 

1908  Apr.  12.0  I'' 
1918  Mar.  11  0 

Berlin  
Greenwich 

6     1  11 
6     2  00 

301.5134 
299  803 

1910.0 
1919  0 

R.  Andersen 
J   M   V  Hansen 

F  

1908  Mar.  23.5)  .. 

Berlin  

6     7  13 

301.2194 

1910.0 

Kris  tense  ri 

Osculation.  .  . 

1809  Apr.  12.0  /.  . 

CELESTIAL   MECHANICS:    LEUSCHNER 


(716)  BERKELEY,  1911  MD. 

Discovered  by  Palisa1  at  Vienna  on  July  30,  1911. 

The  first  preliminary  Elements  A  were  computed  by  Hopfner2  and 
are  based  on  observations  1911,  July  30,  August  17,  and  September  3. 
He  also  gives  an  approximate  ephemeris  for  1911. 

Hopfner's  Elements  A  represented  the  observation  of  1906,  July  16, 
for  (1906  UN1*)  as  follows:  Aa  — lm.4  AS  —  4'.8.  Berberich3  an- 
nounces (1906  UN1*)  is  identical  with  (716)  Berkeley. 

An  ephemeris  is  published  by  Cohn4  for  1912.  An  observation  by 
Palisa5  for  November  4,  1912,  gives  a  correction  to  the  ephemeris, 
Aa  — 2m.7  and  AS  —  8'. 

A  new  set  of  Elements  B  computed  by  Stracke6  is  based  on  obser- 
vations 1911,  August  3,  29,  and  September  23.  These  elements  repre- 
sent the  observation  of  1906,  July  16,  for  (1906  UN1*)  as  follows:  Aa 
+2m.4  and  AS  +4'. 

Stracke's  elements  are  published  and  used  in  B.  J.  1915  and  up  to 
1919  in  Kleine  Planeten. 

An  orbit,  Elements  C,  based  on  three  normal  places  1911,  July  30, 
August  22,  Septemer  17,  was  computed  by  Neubauer7  using  Leusch- 
ner's  Short  Method.8  A  comparison  between  observation  and  compu- 
tation, for  observations  in  1906,  1912,  1914,  of  Elements  C  and 
Stracke's  Elements  B,  showed  that  Elements  C  gave  the  better  repre- 
sentation. Neubauer  also  computes  general  perturbations  by  the 
Bohlin  method,9  with  the  tables  computed  by  Wilson10  for  the  group 
750.11  More  accurate  starting  elements  connecting  several  oppositions 
by  special  perturbations  seem  to  be  desirable. 


REFERENCES 


A.  N.    189,  p.  109. 

A.  N.    189,  p.  244. 

A.  N.    189,  p.  364. 

A.  N.    192,  p.  426. 

A.  N.    193,  p.  62. 

A.  N.    192,  p.  421-423. 

L.  0.  Bull.    No.  301. 

L.  O.  Pub.    vol.  vii,  part  viii. 


'"Formeln  and  Tafeln  zur  gruppen- 
weise  Berechnung  der  Allgemeinen 
Storungen  Benachbarter  Planeten,"  Up- 
sala  1896,  and  "Sur  le  Developement 
des  Perturbations  Planetaires,"  Stock- 
holm 1902. 

10Ast.  lakttagelser  Och  Undersok- 
ningar,  Band  10,  No.  1,  Stockholm. 


TABLE  18.— Elements  (716)— Berkeley  (1911  MD) 


Letter 

Epoch 

M.T. 

M 

, 

fi 

i 

A 

1911  Aug    17  5 

Berlin    . 

120  23     1 

46     3    5 

146  51  41 

8  37  59 

B  

1911  Aug.   18.5  

Berlin  

118    6  10 

48  49     6 

146  57     7 

8  27  43 

C  

1911  Aug    22  4 

Berlin... 

118  25  50 

48  49  15 

146  54  49 

8  23  38 

CELESTIAL   MECHANICS:    LEUSCHNER 
TABLE  IS.— Elements  (716)— Berkeley  (1911  MD}.— Continued 


67 


Letter 

Epoch 

M.T. 

<f> 

u 

Equinox 

Authority 

0        t          * 

, 

A  
B 

1911  Aug.   17.5  
1911  Aug    18  5 

Berlin  
Berlin 

5  28  46 
5     5  17 

753.940 
754  565 

1911.0 
1911  0 

Hopfner 

Stracke 

C  

1911  Aug.   22.4  

Berlin  

5  23  38 

753.7233 

1911.0 

Neubauer 

68  CELESTIAL   MECHANICS:    LEUSCHNER 

(718)  ERIDA,  1911  MS. 

Discovered  by  Palisa1  at  Vienna,  1911,  September  29. 

Preliminary  Elements  A,  and  an  ephemeris  based  on  observations 
1911,  September  29,  October  13,  and  October  28,  are  published  by 
Cohn.2  These  elements  were  used  by  the  B.  J.  1915  and  1916. 

New  Elements  B  were  obtained  by  Strehlow3  from  observations  1914, 
February  28,  March  18,  and  March  29.  The  method  of  the  variation 
of  the  distances  was  utilized  so  that  observations  of  1911  and  1914 
were  well  represented. 

A  correction  of  +1".5  to  /A  was  applied  in  order  to  represent  observa- 
tions of  1904  for  (1904,  OD)  which  is  supposed  to  be  identical  to 
(718)  Erida. 

As  a  test  of  Leuschner's  Short  Method  for  computing  orbits,4  Mundt 5 
has  published  two  sets  of  elements  C  and  D,  which  are  based  on  the 
observations  used  by  Cohn.2 

For  the  purpose  of  comparison,  ephemeris  places  were  computed 
from  Cohn's  Elements  A,  Strehlow's  Elements  B,  and  Mundt's  Ele- 
ments C. 

1917  G.  M.  T.  a                      8 

Oct.  22.5  2h56m3  +17°  09'  Elements  C. 

Oct.  22.5  255.9  +17   07  Elements  B. 

Oct.  22.5  253.5  +1656  Elements  A. 

Dec.    1.5  225.3  +15   39  Elements  C. 

Dec.   1.5  225.1  +15   36  Elements  B. 

Dec.    1.5  222.9  +15   24  Elements  A. 

-Ne  comparison  with  observations  in  1917  has  been  made. 

Strehlow's  Elements  B  have  been  published  and  utilized  by  B.  J. 
(Kleine  Planeten)  since  1915.  An  observation  by  Palisa6  on  1919, 
January  6,  gave  corrections  to  the  ephemeris  as  follows:  Aa — lm.7 
AS— 3'. 

The  object  of  Mundt 's  work  was  to  test  the  possibility  of  deriving 
by  properly  chosen  methods  as  satisfactory  elements  from  a  single 
opposition  as  are  ordinarily  obtained  from  at  least  two  oppositions. 
This  result  was  realized  in  this  case.  His  work  was  duplicated  by 
Miss  Easton5  on  a  slightly  different  plan  of  removing  the  residuals. 
Elements  D. 

REFERENCES 

1  A.  N.    vol.  189,  p.  295.  4  L.  O.  Pub.    vol.  vii. 

3  A.  N.    vol.  192,  p.  421-423.  6  L.  O.  Bull.    No.  302. 

'B.  J.    1917,  p.  36  and  106.  'E.  Z.  der  A.  N.    1919,  No.  561. 


CELESTIAL   MECHANICS:   LEUSCHNER 
TABLE  19.— Elements  (718)  Erida  (1911  MS) 


Letter 

Epoch 

M.T. 

M 

H 

a 

i 

A  

1911  Sept.  29.5  

Berlin  

149     0  40 

169  56  47 

39  22  47 

7     3  55 

B  
C  

1914  Apr.      1.5  
1911  Oct.     13.4  

Berlin  
Berlin  

320  18  15 
156  34  10 

168     8  30 
166  36  12 

39  44  16 
39  42  41 

6  58  13 
6  59     5 

D  

1911  Oct.    13.4  

Berlin  

156  03  25 

166  55  59 

39  43  51 

6  58  49 

Letter 

Epoch 

M.T. 

V 

M 

Equinox 

Authority 

Of* 

r 

A  

1911  Sept.  29.5  

Berlin.... 

12     5  35 

664.65 

1911.0 

F.  Cohn 

B  

1914  Apr.      1.5  

Berlin.... 

11  28  39 

664.412 

1910.0 

Strehlow 

C  

1911  Oct.     13.4  

Berlin.... 

11  19     7 

663.865 

1911.0 

C.  Mundt 

D  

1911  Oct.    13.4  

Berlin.... 

11  20  11 

663.769 

1911.0 

Miss  E.  J.  Easton 

70  CELESTIAL   MECHANICS:    LEUSCHNER 

(884)   PRIAMUS,  1917  CQ. 

Discovered  by  Wolf1  at  Heidelberg  on  September  22,  1917. 

Wilkens  outlines2  his  preliminary  investigation  regarding  the 
motion  of  this  fifth  member  of  the  Trojan  group.  For  this  purpose 
he  utilizes  preliminary  Elements  A,  by  Berberich.  The  libration 
point  is  60°  behind  Jupiter  and  the  planet  oscillates  about  this  point 
in  approximately  150  years.  He  states  that  /*  varies  between 
294".27  and  303".99.  He  also  points  out  that  his  succeeding  work 
will  show  that  Priamus  and  Patroclus  are  diametrically  opposite  in 
the  small  libration  ellipse. 

Klose  applies3  Wilkens'  method4  for  taking  into  account  the  prin- 
cipal perturbations  of  Jupiter,  to  Priamus.  In  this  method  the  prin- 
cipal perturbations  by  Jupiter  are  accounted  for  by  centering  in 
the  Sun  the  mass  of  the  Sun-Jupiter  system.  For  this  study 
Klose  utilizes  Berberich 's  Elements  A,  bringing  them  up  to  mean 
equinox  1925.0,  Elements  B,  and  compares  the  coordinates  computed 
from  Elements  B  plus  special  perturbations  with  those  computed  from 
Elements  C,  which  were  derived  by  Wilkens'  method.  The  differ- 
ence in  the  representation  by  the  two  sets  of  elements  for  observa- 
tions from  1917  October  14  to  1918  December  28,  did  not  exceed  Os.5 
in  right  ascension  and  2"  in  declination.  Klose  concludes  that  Wil- 
kens' method  is  applicable  beyond  this  short  period  for  immediate 
Ephemeris  purposes. 

As  an  example  of  his  method5  of  integrating  the  differential  equa- 
tions for  the  perturbations  in  the  coordinates  for  planets  of  the 
Jupiter  group,  Wilkens6  gives  a  numerical  application  for  Priamus. 
His  results  are  similar  to  those  of  Klose.  Klose7  publishes  a  further 
comparison  between  the  usual  method  of  special  perturbations  due 
to  Jupiter  and  Wilkens'  method.  For  this  purpose  he  compares 
results  gotten  from  Elements  B  and  a  new  set  of  Elements  D, 
derived  by  Wilkens'  method.  He  then  brings  up  Elements  B  with 
perturbations  due  to  Jupiter  up  to  epoch  1918  (Elements  E),  and 
Elements  D  are  brought  forward  to  epoch  1918  by  Wilkens'  method. 
An  ephemeris  is  published  for  the  opposition  1919  for  which  the  per- 
turbations of  Saturn  are  also  taken  into  account. 

In  an  article  on  "Bemerkenswerte  Eigenschaften  der  Bahnen  der 
Planeten  der  Jupitergruppe,"  Wilkens  8  points  out  that  the  ascending 
nodes  of  the  six  Trojan  planets  lie  with  one  exception,  (Patroclus),  in 
the  same  quadrant.  He  also  forms  a  mean  value  of  the  ascending 
nodes  from  certain  planets  of  the  group  and  compares  the  individual 
values  with  these  mean  values.  He  also  refers  to  his  article9  regarding 
the  time  of  maximum  and  minimum  for  the  mean  motion  of  Priamus 


CELESTIAL   MECHANICS:    LEUSCHNER 


71 


and  Patroclus.  He  finds  the  maximum  of  Priamus  (1986.8)  takes 
place  when  the  planet  is  on  a  line  with  and  between  the  point  of 
oscillation  and  the  Sun,  and  the  minimum  for  Patroclus  (1985.8)  when 
the  planet  is  on  the  line  with  and  beyond  the  point  of  oscillation  and 
the  Sun.  This  verified  his  previous  conclusion2  that  the  two  planets 
are  diametrically  opposite  in  their  paths  of  oscillation. 

A  set  of  elements  and  an  ephemeris  are  published  in  Kleine  Planeten 
for  1920.  The  elements  are  probably  Elements  B,  brought  up  to 
epoch  1925.  An  ephemeris  is  published  in  Kleine  Planeten  for  1921. 

In  A.  N.  Vol.  215,  No.  5147,  Wilkens  outlines  his  investigation 
regarding  the  secular  variations  of  the  major  axes  of  the  orbits  for 
the  Trojan  group. 

REFERENCES 


'A.  N.  vol.  205,  p.  141.    M.  N.    78, 
p.  289. 

2  A.  N.  vol.  207,  p.  9. 

8  A.  N.  vol.  207,  p.  183. 


4  A.  N.    vol.  205,  No.  4906. 


'A.  N.  vol.  206,  No.  4937. 

•A.  N.  vol.208,  No.  4984. 

7  A.  N.  vol.  209,  No.  5016. 

8  A.  N.  vol.  215,  No.  5147. 

9  A.  N.  vol.  206,  No.  4945. 


TABLE  20.— Elements  (884)— Priamus  (1917  CQ.) 


Letter 

Epoch 

M.T. 

M 

« 

Q 

i 

A 

1917  Sept.  24  .  5  

Greenwich  .  . 

83  18  55 

329  32  38 

300  41  27 

8  51  26 

B 

1917  Sept.  24  5  

Greenwich.  . 

83  18  55 

329  32  18 

300  48  28 

8  51  28 

C  
D 

1917  Sept.  24.5  
1917  Sept.  24.5  

Greenwich.. 
Greenwich  .  . 

83  46  41 
82  22  47 

329  4  44 
329  51  19 

300  48  28 
300  49  27 

8  51  28 
8  51  24 

E 

1918  Oct.  29.5  

Greenwich  .  . 

115  11  59 

329  45  49 

300  49  27 

8  51  24 

F 

1918  Oct.  29.5  

Greenwich.  . 

115  33  19 

329  24  42 

300  49  27 

8  51  24 

Letter 

Epoch 

M.T. 

V 

ft 

Equinox 

Authority 

A  

1917  Sept.  24.5  

Greenwich  .  . 

o     /       • 

6  46  53 

* 
294.427 

1917.0 

Berberich 

B  
C  

1917  Sept.  24.5  
1917  Sept.  24.5  

Greenwich  .  . 
Greenwich  .  . 

6  46  53 
6  46  54 

294.427 
294.989 

1925.0 
1925.0 

Berberich 
Berberich-Klose 

D 

1917  Sept.  24  5.    .  .    . 

Greenwich  .  . 

7     5  53 

294.427 

1925.0 

Klose 

E 

1918  Oct.    29.5  

Greenwich  .  . 

7     5  59 

294  5850 

1925.0 

Klose 

F  

1918  Oct.    29.5  

Greenwich.  . 

7     7  34 

295.0965 

1925.0 

Klose 

72  CELESTIAL   MECHANICS:    LEUSCHNER 

TABLE  21.— References  for  Observations  of  (884)  Priamus  (1917  CQ.) 


Date 

Place 

Reference 

1917... 

September  22..  . 

Heidelberg  

f  A.N.  205  p.  141,  M.N.  78  p.  289* 
\E.Z.  1917  No.  535 

September  23.  .  . 

Heidelberg  — 

A.N.  205  p.  141;  E.Z.  1917  No.  535 

September  24  ... 

Heidelberg  

A.N.  205  p.  141;  E.Z.  1917  No.  535 

September  25... 

Heidelberg  — 

A.N.  205  p.  141;  E.Z  1917  No.  535 

September  26.  .. 

Heidelberg  

A.N.  205  p.  141;  E.Z.  1917  No.  535 

October        6... 

Wien  

A.N.  207  p.  149 

October       11... 

Wien  

A.N.  207  p.  149 

October       13  ... 

Wien  

A.N.  207  p.  149 

October       16.  .. 

Wien 

A.N.  207  p.  149 

October      21  ... 

Heidelberg  

A.N.  205  p.  239  E.Z.  1917  No.  537 

November    8  ... 

Heidelberg  

A.N.  205  p.  239  E.Z.  1918  No.  537 
A.N.  206  p.  63 

December     4  ... 

Heidelberg  

A.N.  205  p.  279  E.Z.  1918  No.  538 
A.N.  206  p.    63 

1917... 

December     4  ... 

Bergedorf  .  .  .  . 

A.N.  208  p.    39  E.Z.  1918  No.  559 

1918... 

January        2  ... 

Bergedorf  .  .  .  . 

A  N.  208  p.    39  E.Z.  1918  No.  559 

January        3  ... 

Heidelberg  — 

/A.N.  206  p.    63 
\A.N.  206  p.    15 

January      14  ... 

Bergedorf.  .  .  . 

A.N.  208  p.    39  E.Z.  1918  No.  559 

October        5  ... 

Heidelberg  

A.N.  207  p.  239  E.Z.  1918  No.  555 

October      30.  .. 

Heidelberg.  .  .  . 

A.N.  207  p.  283  E.Z.  1918  No.  557 

November  23  ... 

Heidelberg  — 

f  A.N.  208  p.    13  E.Z.  1918  No.  558 
\A.N.  208  p.  167 

1919... 

October      21  ... 

Heidelberg  — 

B.Z.  1919  No.  13  Vol.  1 

1921... 

January      15... 

Heidelberg  

B.Z.  1921  No.  3  Vol.  3 

'Discovery  date. 


CELESTIAL   MECHANICS:    LEUSCHNER 


73 


(911)  AGAMEMNON,  1919  FD. 

This  planet  was  discovered  by  Reinmuth1  at  Heidelberg  on  March 
19,  1919. 

The  nature  of  its  motion  (Jupiter  group)  was  noted  about  the 
same  time  by  Palisa  and  Berberich.2  The  preliminary  elements  A 
by  Berberich  are  published  in  A.  N.  Vol.  208,  p.  332.  A  comparison 
between  computation  and  observation3  for  1919  May  20,  gives 
Aa  =  —4s  AS  =  —2'. 

Elements  A  are  brought  forward  to  mean  equinox  of  1925.0  and 
with  an  ephemeris  are  published  in  Kleine  Planeten  for  1920,  p.  25 
and  p.  47. 

An  observation4  on  March  11,  1920,  gives  a  correction  to  the 
ephemeris  of  +lm-5  in  right  ascension  and  — 18'  in  declination. 

An  ephemeris  for  1921  is  given  in  Kleine  Planeten  for  1921,  p.  20. 

REFERENCES 

1  A.  N.    vol.  208,  p.  231.    Royal  A.  S.      210,  p.  241. 

vol.  80,  p.  405,  406.  8  A.  N.    vol.  210,  p.  244. 

2  A.  N.    vol.  208,  p.  331.    A.  N.    vol.         4B-Z  der  A.  N.    No.  13,  1920. 

Elements— (911)  Agamemnon  (1919  FD.) 
Epoch  M.T.        M.  cu  «  i 

A  1919  Mar.  19.5  Grw.  88°48'19"  78°46'08"  336°55'10"  21°56'50" 

<£  /A        Equinox    Authority  Remarks 

4°55'43"          303".190    1919.0      Berberich.    Preliminary  orbit. 

TABLE  22.— References  for  Observations  of  (911)  Agamemnon  (1919  FD) 


Date 

Place 

Reference 

1919... 

March  19  
April       2  

Konigstuhl.  .  . 
Konigstuhl  .  .  . 

A.N.  208  p.  231* 
A.N.  208  p.  231 

May      20  

Konigstuhl  .  .  . 

A.N.  208  p.  347 

April       5 

Wien 

A.N.  211  p.  430 

April       6  

Wien  

A.N.  211  p.  430 

April     19   .    ... 

Wien     

A.N.  211  p.  430 

April     19 

Wien 

A.N.  211  p.  430 

April     23  

Wien  

A.N.  211  p.  430 

April     25  

Wien     

A.N.  211  p.  430 

April     29 

Wien 

A.N.  211  p.  430 

May        1  

Wien  

A.N.  211  p.  430 

1920.  .  . 

March  11  

Bergedorf  .... 

B.  Z.  1920,  No.  13 

*Discovery  Date. 


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Number  21.  Scientific  abstracting.  By  Gordon  S.  Fulcher.  September,  1921.  Pages 
15.  Price  $0.20. 

Number  22.  The  National  Research  Council.  Its  services  for  mining  and  metallurgy. 
By  Alfred  D.  Flinn.  October,  1921.  Pages  7.  Price  $0.20. 


Number  23.  American  research  chemicals.  By  Clarence  J.  West.  September,  1921. 
Pages  28.  Price  $0.50. 

Number  24.  Organomagnesium  compounds  in  synthetic  chemistry :  a  bibliography  of 
the  Grignard  reaction  1900-1921.  By  Clarence  J.  West  and  Henry  Oilman.  Janu- 
ary, 1922.  Pages  103.  Price  $1.50. 

Number  25.  A  partial  list  of  the  publications  of  the  National  Research  Council  to 
January  i,  1922.  February,  1922.  Pages  15.  Price  $0.25. 

Number  26.  Doctorates  conferred  in  the  sciences  by  American  universities  in  1921. 
Compiled  by  Gallic  Hull  and  Clarence  J.  West.  March,  1922.  .Pages  20.  Price 
$0.20. 

Number  27.  List  of  manuscript  bibliographies  in  geology  and  geography.  Compiled 
by  Homer  P.  Little.  February,  1922.  Pages  17.  Price  $0.25. 

Number  28.  Investment  in  chemical  education  in  the  United  States,  1920-1921.  By 
Clarence  J.  West  and  Gallic  Hull.  March,  1922.  Pages  3.  Price  $0.15. 

Number  29.  Distribution  of  graduate  fellowships  and  scholarships  between  the  arts 
and  sciences.  Compiled  by  Gallic  Hull  and  Clarence  J.  West,  April,  1922.  Pages 
5.  Price  $0.15. 

Number  30.  The  first  report  of  the  committee  on  contact  catalysis.  By  Wilder  D. 
Bancroft,  chairman.  In  collaboration  with  the  other  members  of  the  committee. 
April-July,  1922.  Pages  43.  Price  $0.50. 

Number  31.  The  status  of  "clinical"  psychology.  By  F.  L.  Wells.  January,  1922. 
Pages  12.  Price  $0.20. 

Number  32.  Moments  and  stresses  in  slabs.  By  H.  M.  Westergaard  and  W.  A. 
Slater.  April,  1922.  Pages  124.  Price  $1.00. 

Number  33.  Informational  needs  in  science  and  technology.  By  Charles  L.  Reese. 
May,  1922.  Pages  10.  Price  $0.20. 

Number  34.  Indexing  of  scientific  articles.  By  Gordon  S.  Fulcher.  August,  1922. 
Pages  16.  Price  $0.20. 

Number  35.  American  research  chemicals.  First  revision.  By  Clarence  J.  West. 
May,  1922.  Pages  37.  Price  $0.50. 

Number  36.  List  of  manuscript  bibliographies  in  chemistry  and  chemical  technology. 
By  Clarence  J.  West.  [In  press.] 

Number  37.  Recent  geographical  work  in  Europe.  By  W.  L.  G.  Joerg.  July,  1922. 
Pages  54.  Price  $0.50. 

Number  38.  The  abstracting  and  indexing  of  biological  literature.  J.  R.  Schramm. 
November,  1922.  Pages  14.  Price  $0.25. 

Number  39.  A  national  focus  of  science  and  research.  George  Ellery  Hale. 
November,  1922.  Pages  16.  Price  $0.25. 

Number  40.    The  usefulness  of  analytic  abstracts.    Gordon  S.  Fulcher.     [In  press.] 

Number  41.  List  of  manuscript  bibliographies  in  astronomy,  mathematics,  and 
physics.  Clarence  J.  West  and  Gallic  Hull.  [In  press.] 

Orders,  accompanied  by  remittance,  should  be  addressed  to 

PUBLICATIONS  DEPARTMENT, 
NATIONAL  RESEARCH  COUNCIL 

WASHINGTON,  D.  C. 


The  National  Research  Council 

Membership  and  Organization. — The  National  Research  Council 
is  a  cooperative  organization  of  scientific  men  of  America,  including  also 
a  representation  of  men  of  affairs  interested  in  engineering  and  industry 
and  in  the  "pure"  science  upon  which  the  applied  science  used  in  these 
activities  depends.  Its  membership  is  largely  composed  of  accredited 
representatives  of  about  seventy-five  national  scientific  and  technical 
societies. 

The  Council  was  established  at  the  request  of  the  President  of  the 
United  States,  under  the  Congressional  charter  of  the  National  Academy 
of  Sciences,  to  coordinate  the  research  facilities  of  the  country  for  work  on 
war  problems  involving  scientific  knowledge.  In  1918,  by  Executive  Order, 
it  was  reorganized  as  a  permanent  body.  Although  partly  supported  during 
the  war  period  by  the  government  and  primarily  devoted  at  that  time  to 
its  activities,  the  Council  now  derives  all  of  its  financial  support  from  other 
than  governmental  sources  and  is  entirely  controlled  by  its  own  represeti 
tatively  selected  membership  and  democratically  chosen  officers.  It  main- 
tains, however,  a  close  cooperation  with  government  scientific  bureaus  and 
their  activities. 

Purpose. — The  Council  is  neither  a  large  operating  laboratory  no^ 
a  repository  of  funds  to  be  given  away  to  scattered  scientific  workers  or 
institutions.  It  is  rather  an  organization  which,  while  clearly  recognizing 
the  unique  value  of  individual  work,  hopes  especially  to  bring  together 
scattered  work  and  workers  and  to  assist  in  coordinating  scientific  attack 
in  America  in  any  and  all  lines  of  scientific  activity.  Its  essential  purpose 
is  the  promotion  of  scientific  research  and  of  the  application  and  dissemi- 
nation of  scientific  knowledge  for  the  benefit  of  the  national  strength  and 
well-being. 

Research  Fellowships 

The  Council  maintains,  with  the  financial  assistance  of  the  Rockefeller 
Foundation  and  General  Education  Board — to  the  amount  of  one  million 
dollars,  to  be  expended  during  a  period  of  five  years — two  series  of 
advanced  fellowships. 

Fellowships  in  Physics  and  Chemistry. — Candidates  must  already 
have  made  the  doctor's  degree  or  have  equivalent  qualifications  and  have 
demonstrated  a  high  order  of  ability  in  research.  Address  applications  to 
Secretary,  Fellowships  Board,  National  Research  Council,  Washington,  D.C. 

Fellowships  in  Medicine. — Both  graduates  in  medicine  and  doctors 
of  philosophy  in  one  of  the  sciences  of  medicine,  or  in  physics,  chemistry, 
or  biology  are  eligible  for  these  fellowships.  Address  applications  to 
Chairman,  Division  of  Medical  Sciences,  National  Research  Council, 
Washington,  D.  C. 


"THERE  IS  MORE  UNKNOWN  THAN  KNOWN,"  SAYS  THE 
SCIENTIST,  "BUT  THERE  IS  MUCH  KNOWN  THAT  IS  UN- 
KNOWN BY  MANY,"  SAYS  THE  INFORMATION  SERVICE. 


Knowledge  is  often  hidden  and  must  be  sought' 
in  strange  places.  Without  a  key  to  the  sources 
of  knowledge,  the  seeker  searches  in  vain. 


RESEARCH 

INFORMATION 

SERVICE 

SPECIALIZES  IN  SOURCES 


Its  aim  is  to  aid  research  workers  everywhere;  to  refer  the 
worker  to  the  source  when  available,  when  not,  to  bring  the 
source  to  the  inquirer  by  letter,  abstract,  or  photostat.  From 
its  vantage  point  of  location  and  organization  it  has  unusual 
access  to  international  as  well  as  national  information. 


Its  aim  is  to  aid.     Its  ambition  is  "wider  usefulness. 


THE  RESOURCES  OF  THE  SERVICE  ARE  AT  THE  DISPOSAL 
OF  THOSE  WHO  ARE  INTERESTED  IN  THE  INCREASE  OF 
KNOWLEDGE  AND  THE  FURTHERANCE*  OF  RESEARCH  IN 
THE  NATURAL  SCIENCES  AND  THEIR  TECHNOLOGIES. 


RESEARCH  INFORMATION  SERVICE 

NATIONAL  RESEARCHTCOUNCIL 

WASHINGTON,  D.  C. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


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